In terms of species, the Doctor's greatest enemies are the Daleks, the only foe to have appeared in every Doctor's television era, and responsible for many companion departures and the Last Great Time War that left the Doctor the last of the Time Lords. While their attitude towards their enemies vary between incarnations, the Doctors unambiguously comment that the Daleks are the most evil force in the universe, and show a fear towards them unshared by their other his enemies. The Daleks, meanwhile, openly acknowledge the Doctor as "the Enemy of the Daleks", and by the new series, the idea of facing the Doctor fills them with utter dread, with an entire "intensive care" wing in the Dalek Asylum being devoted the Daleks that survived the Doctor. Complicating matters is how the Doctor had been sent to interfere in the Daleks creation in "Genesis of the Daleks", a Wham Episode of franchise proportion, with his failure to destroy them at their beginning, after being implied to be the very reason they ventured from Skaro at all, forever linking the Doctor to the carnage they spread, a fact that makes them hate the Daleks all the more.
Behind the Daleks are the Cybermen, the Doctor's second most recurring enemies, and responsible for at least two of their regenerations, and even more companion exits than the Daleks. The Doctor has a degree of pity for the Cybermen, both due to their stagnant existence and the fact they house a living being that didn't ask to be made into a Cyberman, but that doesn't stop them from seeing the Cyberman as a corruption of life and perversion of what they view as the very meaning of living. For their part, while they are not as infested as the Daleks due to Emotion Suppression, the Cybermen do recognise the Doctor as an enemy figure that needs to be destroyed.
An honourable mention also goes to the Sontarans, who, thanks to retcons made by the Expended Universe, share the distinction with the Daleks of facing every incarnation of the Doctor, by their thirteenth incarnation, including the War Doctor. In fact, whenever the Doctor lists off their greatest enemies, the Sontarans hold a stronghold in third place, behind only the Daleks and Cybermen. Though, because they are more individualised than them, the Doctors general attitude and relationship with the Sontarans varies on the encounter, as opposed to their usual spats with the Daleks and Cybermen.
In terms of individuals, the Doctor has his old friend, The Master, the polar opposite of the Doctor in almost every respect, with the Master wanting to own the universe that the Doctor is content to just explore. After their childhood friendship came to an end, with the reason varying depending on the source material, the Master made it a priority to either one up or flat-out kill the Doctor, seeing their rivalry as game between them.
Davros, the creator of the Daleks, is the Doctor's exact foil in terms of motivations and the only individual the Doctor has explicitly called his archenemy, with even the extreme pacifist Fifth Doctor being tempted to just shoot Davros and be done with it. Davros, however, sees a kinship between him and the Doctor as fellow scientists, and once even made to broker a friendship between them on his terms, which the Doctor rejected on moral principle, causing Davros to intensify their animosity by declaring war on the Time Lords themselves, which was only delayed when the Seventh Doctor tricked him into destroying Skaro. Being a death's door almost caused Davros to settle his feud with the Twelfth Doctor, until it turned it to be a ploy to extend his own life.
Following the Last Great Time War, Rassilon, the founder of Time Lord society resurrected for the war, became another major thorn in the Doctor's side, being one of the reasons the Time Lord embraced the war and become warmongers, despite the Doctor's efforts to keep them peaceful. Once he made it back to Gallifrey, the first thing the Doctor did was oust Rassilon as President and banish him in penance for his war crimes.
However, in both the television series and the expanded media, each Doctor has at least one personal rival of their own, either by having a personal beef with that specific Doctor, or by being their most recurring enemy:
First Doctor
The Daleks were his most recurrent foe, first appearing in the show's second serial and quickly becoming staples of his era, to the point where the longest ever Classic Series serial was a "Doctor vs. Daleks" story. Each of the Daleks' appearances turned into a Wham Episode by shaking up the cast, including them being responsible, both indirectly and directly, for the first companion deaths in the franchise. Their ruthlessness was also a key part of the Doctor's own Character Development into a more altruistic figure, with the Daleks being the first villains that the Doctor went out of his way to oppose when they first returned for a rematch.
In the Daleks' last three television serials, they are lead by a Black Dalek, which the expanded universe would retcon into being the same Dalek, called the Black Dalek Leader, who was also made into the singular Black Dalek for the rest of the Classic Series until the Seventh Doctor destroyed it. The Black Dalek Leader would thus be retroactively made an Unknown Rival for the First Doctor, acting as the Big Bad and Greater-Scope Villain for the First Doctor's Dalek stories, even though they only meet face-to-face briefly in "The Daleks' Master Plan".
The Meddling Monk is the only individual enemy that the First Doctor faces more than once on-screen and the only member of his own species that he clashes with. He has a philosophy towards time travel that is directly at odds with the Doctor's own, interfering for his own amusement, which sees the Doctor twice cripple the Monk's TARDIS in an attempt to curtail his facilities, leading to the Monk seeking revenge on him. However, both because he's treated as a Harmless Villain in his first appearance and playing second fiddle to the Daleks in his second, the Doctor does not show much scorn towards the Monk beyond some annoyance.
The Doctor's also got quite the off-screen history with the Celestial Toymaker, his most powerful individual enemy, despite his only having one appearance in the television show. Expanded media picked up on this, and the Toymaker soon become an indispensable part of the Doctor's backstory, where it was revealed that he was responsible for the infamous scandal that got the Doctor expelled from the Academy after an ill-thought out expedition to the Toymaker's realm with two of the Doctor's friends resulted in the death of one and the Toymaker possessing the body of the other, retroactively making their one on-screen battle very personal for the Doctor.
Second Doctor:
The Cybermen fought the Second Doctor more often than any other foe at four serials, and were also indirectly responsible for his previous incarnation regenerating, thus having a hand in the Second Doctor's "origin". In fact, it was during the Second Doctor's era that the Cybermen experienced their own "Dalekmania", resulting in a large array of Expanded Media pitting them against the Second Doctor, with him being seen as their primary enemy after they were Put on a Bus in the 1970s, due to the new showrunners not particularly liking them.
The Great Intelligence was also set up as an arch-enemy for the Second Doctor, but a falling out between the character's creators and the production staff meant that it never fully materialized until the Eleventh Doctor's final season. That being said, the Intelligence is forever linked to the Second Doctor due to the popularity of its two serials, in no small part to the second setting up the friendship between the Doctors and the Lethbridge-Stewart family, and the same licencing agreements that resulted in it being Put on the Bus leaving the Intelligence to hold "the Cosmic Hobo" in particular contempt.
During his TV Comics run, while the Cybermen remained a consistent threat during John and Gillian's time with the Doctor, they become overshadowed by the Quarks by the time Jamie re-joined the TARDIS, as the comics' publishers believed the Quarks would replace the Daleks in popularity after "The Dominators" aired. While their assumption proved false on a television front, the Quarks did end up becoming a prominent enemy to the Doctor in the weekly comics, with them swearing revenge on the Second Doctor in particular after he foiled their first invasion of Earth.
Third Doctor:
The UNIT era Master was the Doctor's most recurring enemy, often turning out to be behind the incidents the Doctor and UNIT had to deal with, and his most personal enemy on account of the two being former friends, and even retaining their banter as Friendly Enemies, with the Doctor even taking time to visit the Master in prison. While the Doctor initially saw him an "unimaginative plodder" that only caused trouble, he eventually came to see the Master as the "personification of evil", and held no qualms about leaving the Tremas Master to the mercy of an ambush in the Death Zone of Gallifrey, albeit under the assumption that the Master had set up the ambush.
The Daleks are a clear second for the most recurring enemy of the Third Doctor, with two six-part serials unofficially forming a twelve-part story on par with "The Daleks' Master Plan" by having the Doctor stop their most recent ambitious scheme to take over the universe by unleashing a Dalek army on a Galactic Alliance weakened by the Master as part of an alliance between the two villains. Not only did the Keller Machine reveal his fear of the Daleks in a deviation from his usual fearless persona, but the Doctor also sidesteps his pacifism by taking "great satisfaction" in destroying them, and also going for the kill shot when dealing with their Ogron minions. Evidently, while the Master brings out his cunning, it is the Daleks that bring out the worst in the Third Doctor.
Fourth Doctor:
The Fourth Doctor holds a particular grudge against Davros, as he is the only incarnation to face the Daleks' creator twice on-screen, with the Mad Kaled being the first villain to push the Fourth Doctor to his limits, partially because "Genesis of the Daleks" was a particularly bleak story in an otherwise Lighter and Softer season, with "Destiny of the Daleks" deepening the already existing enmity between them after Davros was set up as the complete antithesis to everything the Doctor stood for, with the Doctor taking great glee in personally freezing Davros as an imprisonment. According to the Expanded Universe, Davros would have nightmares about the Fourth Doctor in particular, holding him accountable for ruining his life.
The Master holds the spot of the most recurring villain in the Fourth Doctor era at three serials, starting with the Decayed Master, transformed to accommodate the Darker and Edgier nature of the late 1970s era, with his next story having him "regenerate" into the Tremas Master to pave way for the Denser and Wackier 1980s era, but whose machinations would set in motion the chain of events leading to the Fourth Doctor's regeneration. In the Big Finish Doctor Who audio dramas, the Decayed Master returns to once again be his number one recurring foe, with the nuances of the relationship explored in furfur detail.
Fifth Doctor:
The Tremas Master was the most recurring foe the Fifth Doctor faced over the course of his tenure, with the fifth incarnation also holding second place in the record for most on-screen battles with the Master after the Third Doctor. While they retain some of their Friendly Enemy relationship, the Tremas Master made it his mission to kill the Doctor from his first to last chronological stories, with his machinations indirectly forcing the Doctor to kill his companion Kamelion, resulting in the Doctor standing by as he thought the Master burned to death, content to put his pacifism aside to see his biggest enemy vanquished.
The Fifth Doctor had the biggest animosity with the Cyberman than his other incarnations, due to their involvement with Adric's death in Earthshock, which would remain Five's greatest regret until his final breaths. He was beside himself with fury when he realised he was partially responsible for the genesis of the Cybermen in the Spare Parts audio drama, and was equally enraged when, against all the odds, he bumped into the same Cyber-Leader from Earthshock in the Conversion audio drama. Long story short, when Five goes against the Cyberman, It's Personal.
The Fifth Doctor also found himself periodically going against the Mara when it takes his best friend Tegan as a host, with the Mara's very nature as a personification of evil clashing with the especially good-natured Fifth Doctor, and his battles against the Mara forcing him to draw on willpower he seldom finds himself using. In the audios, the Mara moves from Tegan to using the Doctor himself as a host, cementing its status as one of the Fifth Doctor's most personal enemies.
Sixth Doctor:
The Valeyard is the Doctor's literal darker half and the one who orchestrated a series-long gambit to steal the Sixth Doctors regenerations. The Doctor Who New Adventures would explicitly link the Valeyard to the Sixth Doctor specifically due to his more violent nature, and the Valeyard would ultimately become the Final Boss for the Sixth Doctor when his endgame necessitated the Doctor to regenerate to undue his masterplan.
Including their one on-screen battle, the Sixth Doctor has had the most encounters with Davros than any of his other incarnations thanks to the Big Finish Doctor Who audio dramas, which go on to explore the deeper relationship between the Doctor and Davros, including a Villain Episode that has the pair forced to work together, during which Davros even admits that the Doctor is the closest person he has to a friend, but the Doctor isn't having any of it, but does later acknowledge some sympathy to Davros after a "Freaky Friday" Flip.
Seventh Doctor
Despite showing up only once on-screen, Fenric serves as the Greater-Scope Villain for the entire Seventh Doctor era, and the pair's confrontation in the penultimate story of the Classic Series serves to wrap up Season 25 and 26's focus on the Doctor as a more cunning and manipulative figure bearing dark secrets behind his chipper exterior.
The Eternal Death saw the Seventh Doctor has her main opponent, since he served her fellow Eternal Time as Time's Champion, and held an ire over him for managing to win a bet with her. Death would even take on the Monk as her Champion to spite the Doctor, and also appeared to him when he suffered a one-sided heart attack just to taunt him about his upcoming regeneration. After Death sabotaged the Doctor's attempt to save the Master from himself to make him Death's new Champion, the Doctor retaliated by saving a suicidal tramp just to "get one over on Death".
Eighth Doctor
The Master, who remains the only enemy to tackle the Eighth Doctor in all forms of media he appeared in; the telemovie, the comics, the books and the audios.
After it's machinations resulted in the deaths of his companions, Lucie and Tamsin, and his great-grandson, Alex, the Eighth Doctor came to loathe the Dalek Time Controller, with their enmity growing during the Dark Eyes sage, where the Time Controller continued to hound the Doctor and his new companion, Molly, as part of an alliance with Kotris to destroy the Time Lords. Even after Kotris was killed, the Time Controller continued with the plan, allying instead with the Reborn Master, until it was absorbed into the Eminence.
War Doctor
Since his amenity with the Daleks is treated more along the lines of them being the opposing faction of a war than the personal conflict it was previously, the War Doctor's main enemy is instead Rassilon, as, unlike the Daleks, Rassilon is on the Doctor's side of the war and the Doctor can't strike Rassilon down like he does with the Daleks. Eventually, Rassilon proves the worst of two evils, and the Doctor takes drastic action to stop the war.
The Dalek Time Strategist, as the replacement of the Dalek Time Controller as the Dalek Empire's second-in-command, was the War Doctor's main opponent on the battlefields of the Time War, with the Doctor often being sent to sabotage the Time Strategist's schemes.
Ninth Doctor
The Daleks as as whole count, being the direct representation of his past as the sole survivor of the Last Great Time War and the blurry lines between himself and his enemies, but the Dalek Emperor in particular is this for his role in the Doctor's war trauma and attempt to goad the Doctor into being a killer like the Daleks.
The Slitheen family are his most recurring enemies, with one their numbers, Blon Fel-Fotch Passameer-Day Slitheen, making a comeback to challenge the Doctor's modus operandi, particularly how the ninth incarnation doesn't stick around long and rarely has to see the consequences of his actions, which she sees herself as the living embodiment of.
Tenth Doctor
The Cult of Skaro are the Tenth Doctor's most recurrent foes, popping up for an arc Once a Season, and serve as lynchpins in his relationship and history with his companions, as well as playing on his guilt from the Last Great Time War. Their first appearance resulted in the Doctor losing Rose, with their second appearance having him skirting the Rage Breaking Point due his fury that "they survive[d] while [he lost] everything", but he still extends a helping hand to the last surviving Cultist, Dalek Caan, after seeing the others being killed, only for Caan to use this moment of mercy to escape. Caan would then return with Davros, having rescued him from the Time War at the cost of his sanity, but his insanity opens his mind to the flaws of the Daleks and he manoeuvres events to destroy Davros' New Dalek Empire, which includes sacrificing himself and the Doctor's best friend, Donna, but the Doctor is able to save Donna at the cost of her losing her memories of him, resulting in the Doctor spiralling into a desperation that culminates in the Time Lord Victorious Crisis Crossover event, securing the legacy of the Cult of Skaro as the Tenth Doctor's biggest enemies.
The Saxon Master is the direct Distaff Counterpart to the Tenth Doctor and acts as the bridge between his present-day self and his guilt as the sole survivor of the Time War, with the Doctor trying to reach out to him so they can rekindle their friendship and just enjoy being the last of the Time Lords together, but the Master, rendered more insane than ever, repeatedly reusing the hand of friendship in favour of subjecting the Doctor and his friends to continuous humiliations out of sheer spite.
Eleventh Doctor
The Silence, as lead by Madame Kovarian, were the Eleventh Doctor's most persistent enemies, masterminding the Myth Arc stretching across his entire tenure as they attempt to prevent him reaching Trenzalore to answer the First Question and bring the Time Lord back to restart the Time War, with their machinations ranging from blowing up the TARDIS to kidnaping the Doctor's best friend, Amy, during her pregnancy and molding the child into becoming their assassin, though she rebels and marries the Doctor instead. When the Doctor finally does reach Trenzalore, it transpires that Kovarian was dealt with off-screen after the Silence's true leader, Tasha Lem, learnt that blowing up the TARDIS had allowed the Time Lords to get a foot back into the universe, thus being responsible for the mess she was trying the avoid, which the Doctor calls a "Destiny Trap". He then joins forces with the Silence to fight off the Siege of Trenzalore.
The Great Intelligence is reshaped into the Doctor's Evil Counterpart by how it takes on children as "companions" to enact it's schemes as they grow up, as a parallel to Eleven's meeting with Amy, River and Clara as children. However, it spends much of his time an Unknown Rival to the Eleventh Doctor, who doesn't seem to fully remember him at first and isn't even aware some of the schemes he's thwarting are the Intelligence's, but, by the Series 7 finale, the Great Intelligence establishes itself as one of the Doctor's most diabolical foes with its plot to undo every victory the Doctor achieved by entering his timestream, until it was foiled by Clara.
The Eleventh Doctor also had it bad with the New Dalek Paradigm, seeing them as a direct result of his failure to stop the Dalek race rebirthing itself and holding himself accountable for the destruction they reap. Interestingly, due to the politics of the show's production and the initial backlash against the Dalek redesign, the Eleventh Doctor is the only incarnation that the New Dalek Paradigm faces in any media, even serving as his Final Boss during the Fall of the Eleven in the Siege of Trenzalore.
Twelfth Doctor
Missy is his most recurrent opponent and highlights his era's focus on him questioning his own morality. Although he mainly trades blows with Missy in the television series, the Expanded Universe has the Twelfth Doctor face the UNIT era Master in Doctor Who Magazine, and Titan Comics sees him in a "Multi-Master event" that includes Missy, the Decayed Master, the Bruce Master and the Saxon Master, the last of whom he faced in an onscreen "Multi-Master event" in 2017. During his final Story Arc, the Doctor appoints himself Missy's warden when she is finally imprisoned for her crimes, and seizes the oppurtunity to try and redeem her so they can be friends again, with Missy also deciding to go "cold turkey" because she wants to be friends again too, only for the process to be undone by the Saxon Master's meddling, until Missy kills him to join the Doctor in battle and he in turn fatal wounds her out of spite.
The Cybermen roughly bookend his tenure by appearing in his first and last season finales, both times serving as underlings for an incarnation of the Master; first Missy, and then the Saxon Master. The Cybermen remain his runner up threat even without the Master, showing up Once a Season even if just in cameo throughout his tenure, even during his year off television in Doctor Who: Supremacy of the Cybermen, and are ultimately responsible his regeneration.
Thirteenth Doctor
The Spy Master is the Thirteenth Doctor's most recurring and personal enemy, with his repeated schemes involving her and the revelation of the Timeless Child, culminating in him stealing her body in order to truly leave his past behind and become the Doctor, albeit as a corruption of the name. After he is returned to his dying body, the Master fatal wounds the Doctor out of spite as he dies, resulting in her regenerating.
T'zim-Sha qualifies as one to both the Doctor and Graham, given the role he played in the death of Graham's wife Grace, who was shaping up to become one of the Doctor's new companions. T'zim-Sha also served as Thirteen's Starter Villain, with his return in her first finale cementing him as the Big Bad of her first Myth Arc concerning she and Team TARDIS becoming "the Fam", with his defeat also bringing peace to Graham and Ryan over Grace's death.
Captain John Hart to Jack Harkness, though the two act more as Friendly Enemies in their Distaff Counterpart relationship, helped by the fact that they had had a fling while working for the Time Agency together.
The Trickster is Sarah Jane's ultimate enemy, having devised many plots to remove her from the equation and allow Earth to succumb to chaos, often requiring her to make hard sacrifices to stop him. Next to him are Mrs. Wormwood, the creator of Sarah Jane's adopted son, who makes a claim to being his "real" mother to drive a edge between them, and the Slitheen family, the most recurring enemies in Sarah Jane's Rogues Gallery.
Inspector Drake was this to K9's team, until he was Reassigned to Antarctica and replaced by Inspector Thorne, who become an even more hated adversary even before he revealed himself as an advocate for K9's true arch-foes; the Korven.
