John Adams often believing himself to be this in his unceasing push for Congress to vote for independence. The problem is that he's obnoxious and disliked by near every other delegate regardless of which side of the vote they're on — Benjamin Franklin calls him out on it because Adams' attitude demeans the accomplishments of the other delegates.
John Hancock feels this way as President of the Continental Congress as the antics of its delegates drives him crazy and keep stopping the proceedings for sometimes stupid reasons. His reactions are usually either to let out his frustration by Suddenly Shouting at the target of his ire or conclude that I Need a Freaking Drink.
Arsenic and Old Lace has an entire crazy family, the Brewsters. There are only two characters that know what's going on and are not insane, Mortimer and the unrelated Dr Einstein. (No, not that one.) And Dr. Einstein is a criminal accomplice in way over his head, and a little crazy, too. So Mortimer ends up literally the Only Sane Man. He's ecstatic when he learns he's not related to the crazy people.
Of the handful of characters in Be More Chill that know what the Squip is and what it does, Christine is the only one to realize, right off the bat, that maybe putting a tiny supercomputer into your brain and and having it instruct you on how to be cool is, perhaps, a bad idea. (Of the people who know about the Squip, Christine's also the only girl. Women Are Wiser, perhaps?)
Underling in The Drowsy Chaperone is easily the most sane character in the show: the bride convinces her fiance to cheat on her with herself, the groom goes rollerblading blindfolded after being told that tap dance is too dangerous, the Chaperone poses as the bride in order to be seduced, Tottendale has the memory of a goldfish, Kitty uses fake telepathy to trick Feldzieg into marrying her (and it works), and so on. Even the Man in the Chair gets in on the insanity by joining Janet in Bride's Lament. Underling, by contrast, is a Servile Snarker who is barely able to keep the wedding together.
Leon Tolchinsky from Fools, the only character not to be infected with a curse of stupidity.
Similarly, in Hamilton, Aaron Burr, Alexander Hamilton, and Thomas Jefferson each see themselves as this. George Washington actually is. (Although James Madison is a close second simply by way of having to be extremely serious to act as Jefferson's straight man.)
Hamlet's Horatio is the only person in Elsinore who isn't caught up in the emotional turmoil surrounding Hamlet's supposed madness, as lampshaded by Hamlet himself: "Give me that man that is not passion's slave, and I will wear him in my heart's core, ay, in my heart of heart." As a result, he's the only one alive at the end.
In one parody of Hamlet, Horatio must relate the story of the play to two policemen who are checking to see if they need to press any homicide charges; Horatio himself states that he is now stuck with the exceptionally depressing job of arranging the funerals for and burying all of his friends and their families on his own.
It's from Top Ten Shakespeare Stories by Terry Deary, if anyone's interested.
Marcus Andronicus in Titus Andronicus, at least for a while - by the end even he seems to be losing it.
In H.M.S. Pinafore the role falls to Dick Deadeye. Of course, since he's ugly, hunchbacked and named Dick Deadeye no-one listens to him, even when he's making common-sensical statements like "It's a queer world". In fact, at one point he manages to turn the rest of the crew's opinion right around by agreeing with them.
In Next to Normal, we have the bipolar Diana, the angsty perfectionist Natalie, Natalie's goofy stoner boyfriend Henry, the rebellious, intimidating, has-been-dead-for-years Gabe, Dr. Madden, the terrifying therapist, and... Dan.
Ines in No Exit is a dark take on this trope; being a Card-Carrying Villain, she's not bogged down by attempting to act innocent to the others and is the first to realize that they're all in hell, there for very good reasons, and are meant to torture each other.
Peer Gynt has a moment in the fourth act of the play where the title character ends up in a madhouse, and ends up the Only Sane Man, given that even the keeper of the asylum is slipping.
Benvolio in Romeo and Juliet. Other characters even seem to be aware of it. At two different points, an authority figure asks him to explain what happened, trusting he will be able to say so accurately.
Doc in West Side Story is a more dramatic version of this — he sees that the gang wars are ultimately meaningless, sees that Tony and Maria's relationship is going to end in disaster, and sees that absolutely nobody will listen to him.
In the first act of A Very Potter Musical, Hermione frantically tries to convince the other characters that the House Tournament is dangerous, but their reaction is pretty well summed up by Harry's line: "Duh, Hermione, I'm the Boy Who Lived, not Died". Later on, in the second act (after Hermione has been proved right), Harry gets a turn: Voldemort has returned, but whenever he tries to talk about it with someone, they change the subject to their love lives.
