Devil May Cry / Awesome Bosses

Any time Dante, Vergil or Nero crosses swords with another human-sized, humanoid enemy, including each other, the resulting fight is guaranteed to be awesome.

Devil May Cry

The first fight with Nelo Angelo (a.k.a. Vergil) in the first game. His introduction is badass as all hell, and after Phantom, a down-and-dirty sword fight is just the thing to cleanse the system.
Dante: This stinking hole was the last place I ever expected to find anybody with some guts.
Every one of the Nelo fights, actually. Most DMC bosses that you face like to do patterns on you. Nelo fights like you, can do moves like you, and if you are not at the very top of your game, he will hand you your ass. Especially during the final showdown with him, where he brings those swords into play.

The second duel with Vergil deserves special mention, because you and your opponent are essentially equally matched. Both the player and the boss have two weapons they can swap between to create combos, both have ranged attacks, both have Style-based abilities and both have access to the Devil Trigger for a temporary boost in stats and some health regeneration. But because Vergil (on most difficulties) does more damage and has more health, it's up to Dante (i.e. the player) to outwit and have better reflexes than his rival.
Beowulf. His undodgeable homing projectiles if you get too far away means you are forced into a brutal close-combat slugfest with a brute many times your size who is still fast and capable of instant-death-radius explosions and lots of other attacks.

The first fight with Dante is not all that hard in the early difficulty levels, especially when it is used as a tutorial level, but in Dante Must Die mode, this fight ramps up completely, and it becomes quite challenging and fun, and it is just the first boss of the game. Granted, you've already fought all the bosses at this point, but this beginning boss fight in this mode really sets the stage for the rest of the difficulty mode.
The second time you fight Dante is a memorable, albeit insanely challenging, fight. The first time you fight Dante he's holding back but when the second fight rolls around, all the chips are on the table. Expect to have numerous sword clashes with the Devil Hunter and many instances of Shoot the Bullet (but be wary of your health because this is no cakewalk; manage your Devil Trigger carefully to both rejuvenate health and deal out additional damage). The final battle against Sanctus Diabolica is kind of deflating compared to the epic struggle against Dante at only the midpoint of the game.
The fight against Angelo Credo. Think an Alto Angelo but up to eleven. What's best is being able to throw his lances back at him and open him up to a Devil Bringer attack. Unlike most bosses, Credo is only fought once in the story (and while the battle against Dante does occur twice, it's only the second time that it really gets your blood going, unless you're playing on a higher difficulty at which point both fights can be pretty cool).

The game may not have found its niche for everyone but DmC: Devil May Cry has a couple of good fights. The final battle against Vergil involves some impressive swordplay and Nephilim magic between the two brothers.
The fight against the disembodied, digitized head of Bob Barbas is full of surreal, trippy Cyberspace and even drags you into fake news reports.

Cavaliere Angelo is a fun boss to parry against, and he can serve as a target practice for the idea that bosses can be parried and staggered in the first place. Mission 11 sells his boss fight well by having Dante mock him if you can consistently parry his moves. The boss's knight theme and moves are also compared to Angelo Credo, someone who's already considered a memorable boss from 4.
Both fights against Vergil. With Dante, his fight mirrors his battles against Vergil in 3. With Nero, it shows his determination as he unlocks his Devil Trigger and beats Vergil with it.
For the Special Edition when playing as Vergil, you will fight Dante himself twice as the final boss instead. The first time is a warm up and teaser, getting you used to being on the receiving end of the Legendary Devil Hunter. His second fight though has him going all out, throwing everything he has at you.