The Adventures of Tintin: Professor Calculus in Tintin: Red Rackham's Treasure. In all the other books, his hearing problem leads to simple, albeit comedic, misunderstandings. In Red Rackham's Treasure, though, it's so bad that he's completely off in his own little world. Like all good examples of this trope, he even spouts random non-sequiturs out of the blue.
Ambush Bug: Hello, room service? Send up a plot and three pages of dialogue right away! The weekly grind is tearin' me apart! Fifty-two!!!
Jughead Jones from Archie Comics with his big eating, his girl-hating, and his oddball dress and trademark crown. His oddness tends to vary from barely noticeable to completely insane Depending on the Writer, but he's always the weirdest of the group.
One issue had a rather sad take on this where Jughead only played the part of the oddball to garner laughs as a means of coping with his homelife: his (implied to be single in this issue) mother was very sick and Jughead would fall into a state of depression if he wasn't making people laugh. It ends with the whole gang, even Reggie Mantle, showing up at his house to help him with the housework and provide moral support.
The One-Scene Wonder (even though played by no actual actor) of the British gardener in Asterix in Britain. Trespass on his lawn and he will stab you.
Squareonthehypotenus, who goes absolutely insane in his attempts to raise the Mansion of the Gods.
Norbert Sykes, a.k.a. The Badger, is a prime example. The author tended to vacillate between presenting Badger's mental illness as serious or just an excuse to make him goofy, but he tends to be full of non-sequiturs even at his best, often getting into barfights. He seems to be aware of his own state, though, cartwheeling out the door in one issue while singing "I'm coocoo for cocoa puffs!"
Quite a few of the characters in Bone will occasionally lapse into Cloudcuckoolander territory, but none moreso than Smiley Bone, whose unique worldview makes sense only to him, and who gets quite a few Crazy Awesome moments during the course of the comic.
Rat Creature 1: This is insane! Rat Creature 2: It's stupid! Smiley: Hey! Nothing we've done so far has been un-stupid, and we're still alive, aren't we? Rat Creature 1: I can't really argue with that, but I feel like I should.
The two Stupid, Stupid Rat Creatures also have strong tendencies towards this trope — of the two, the stupider, quiche-obsessed one (generally depicted with brown fur in the color versions) is the most consistent practitioner of the trope, but the smarter, more aggressive one (usually depicted with purple fur, though the colorist occasionally gets confused) has more than his fair share of moments as well.
The Frenchman and The Female in The Boys. Apparently, they understand each other's private moon logic quite well. Well enough to play Monopoly on a Cluedo board, anyway.
Doomguy: "WHO'S A MAN AND A HALF? I'M A MAN AND A HALF!"
In his defense he was buzzing on a berserker pack when he said that, but he's pretty damn strange throughout the whole comic.
Mento of Doom Patrol fame sometimes qualifies as this when wearing the psionic helmet that gives him his powers. It enhances his mind in many ways, but the consequential increased mental activity makes it difficult for him to concentrate. When the helmet malfunctions, it make him fairly eccentric and at one point gave him cancer and dementia.
Grant Morrison's run has a villainous example in the form of The Brotherhood of Dada, who believe that good and evil are concepts of an outmoded age and they must simply do things because they can. Their first major "performance" involves their leader, Mr. Nobody, throwing a dead chicken to the ground and saying they've conquered the world, another member turning a gendarme into a toilet filled with flowers, and the use of a magical painting to trap all of Paris.
Bob Burden's Flaming Carrot, a blue collar surrealist superhero who suffered brain damage from reading 5,000 comic books in one sitting, wears flippers constantly (in case he has to swim) and fights crime because he "needed the exercise."
Airtight from G.I. Joe is described as that weird kid nobody wanted to be around, grown up even weirder. This guy keeps scorpions as pets and eats peanut butter and tomato sandwiches, you guys.
Don't knock them until you've tried them. Just make sure you toast the bread.
Maps from Gotham Academy is the type to fall in love with a grapple gun.
Nathanael Beauregard in The Grievous Journey of Ichabod Azrael - when he falls on another man, his first act is to reassure him that he isn't up to anything gay. He also thinks that limbo is a plot by Native Americans to double-kill him.
I Luv Halloween - Finch's psychotic little sister, "Moochie", whose a mix of both cloud-cuckoo lander and ax crazy. In volume 2, she goes on a zombie-laden quest to hunt for the "King of the Chonklit Monkeys" whom she believes live within peoples' bowels and take all the Halloween candy and replace it with their poo (she ends up being right in the end). She also disembowls an obese woman and rips her face off with her teeth, extracts molars from the mouths of the dead and the living because she was dressed as the tooth-fairy for Halloween, believes a still-born fetus she finds is her sister, and cuts off a man's buttocks and spanks it as victory for exorcising the chonklit monkey king. Moochie: "Is the much oompah oompah victory parade! Spanked his brown bottom we did! Is trick or treaties for everyone! Yay!
The eponymous Lenore the Cute Little Dead Girl of Roman Dirge's comic is a rather dark take on the Cloudcuckoolander, as her inattentiveness, tenuous grasp on reality, and near-nonexistant understanding of the concept of mortality leads her to frequently inadvertently cause the deaths of the people and animals she deals with. She could be considered Ax-Crazy, but she's not truly insane, and usually doesn't intentionally mean to cause harm.
Not Pinkie Pie (for a change), but Rainbow Dash in the second story-arc of My Little Pony: Friendship is Magic (IDW), who suggests using an enlarging ray on her (so she could transport an entire army to the moon all by herself) and a time machine to rescue Rarity.
Planetary: "... the man trying hard not to hump your TV is The Drummer."
"First name The, second name Drummer." "You'll regret being so damn abusive when the electric UFO gods transphase in from Dimension Ten to appoint me Manager of the Universe.... I said that out loud, didn't I?"
Max from Sam & Max: Freelance Police qualifies big time. His capacity to distinguish between fantasy and reality is directly rooted in Sam's capacity to remind him what the difference is.
According to Word of God, Max is actually a powerful psychic multi-dimensional hive mind whose thoughts and memories are consistent across all time and all the universes, and the hallucinations we see when looking through his eyes in The Devil's Playhouse are reflections of the adventures other versions of himself are going on in the past, the future and other worlds. This probably explains why he qualifies so much for this trope, but could just as easily have been the creator trolling his fanbase.
Delirium from The Sandman sometimes comes close to this trope — since she's the Anthropomorphic Personification of insanity, it's probably reasonable to assume that she is genuinely crazy (insofar as this means anything regarding one of the Endless), but nonetheless she does have at least one moment during the series where she pulls herself together and becomes briefly 'sane', though it's made clear that she finds it very difficult to do this. It has been implied that she's not just the incarnation of insanity, but the incarnation of ultimate, painful, sanity-breaking clarity. She also has a few other moments in which she seems to become temporarily slightly more lucid, and comes out with a very perceptive or useful comment before reverting to her usual chaotic self.
Shivering Jemmy also qualifies. She's an incarnation of Chaos (but not like that), after all.
For a more mortal and habitually horrible example, see the treatment of Doctor Destiny in the same series; an insane supervillain with control over dream realities, and just enough clarity to use that power to his own advantage.
The characters in Scott Pilgrim tend to run in and out of this sometimes, but most famously is Scott himself, with such winning lines as:
Stacy Pilgrim: Maybe you should start thinking about the future, Scott. Scott: The future? Like, with jet packs?
Wallace: Scott. It's time to get serious. Break out the L word. Scott: ...Lesbian? Wallace: No, Scott, the other L word. Scott: ...Lesbians?
And the follow-up joke:
Joseph: Have you used the L word yet? Scott: Why is everyone obsessed with lesbians!?
Charmy Bee from Sonic the Hedgehog (Archie Comics) was not originally portrayed this way. Due to the original English translation of Knuckles Chaotix claiming that Charmy was 16 years old, Archie presented him as a Rebel Prince with an Arranged Marriage. When Sega corrected them that he was supposed to actually be 6 years old, Archie solved the discrepancy... by having Charmy suffer brain damage and mental trauma inside one of Eggman's machines, causing him to revert to the mental level and personality of a hyperactive child.
Supergirl antagonist Bizarrogirl, by virtue of being a Bizarro, has a weird, warped mindset. She kidnaps a bunch of strangers because she wants to throw a party but she demands them to remain silent. Later, when Supergirl punches her across the city and she crashes into a park, Bizarrogirl starts to pick flowers, sing and dance.
Ragdoll. The self-castrating, dead-friend-stuffing, sister-fancying, weird-phrase-spouting, limb-contorting freaky-pants of Gail Simone's fantastic Secret Six series. He's quite possibly the only person in the DCU who can make The Joker appear sane by comparison.
Ragdoll: I'm buying a monkey house and a variety of little monkey outfits.
"I was thinking what it would be like to be abandoned and tortured and abused and forgotten. When your life is so worthless that your only degraded value to anyone is when your pain gives them amusement, and the person entrusted to care for you sees you as more disposable than used tissue... but then I thought... 'I wonder what it's like to f*** a butterfly.'"
From the same team, King Shark is shaping up into one of these quite nicely.
King Shark: I got a unique problem. I like to eat people with eating disorders.
Spider-Man: Mary Jane was like this in her earlier appearances. Readers eventually find out there was some Stepford Smiling going on and in the modern era her character is about 100 times more grounded (still a fun character, just not bat crap crazy). Earlier appearances of Aunt May also indicated that she lived in Cloud Cuckooland (the joke being she was senile). Like MJ, she's since mellowed out a lot, creating some Early Installment Weirdness for readers who go back and read collections of the old trades.
As far as Spidey villains go there's White Rabbit. If the Alice in Wonderland theme weren't a tip off then the fact her first villainous plan was to rob fast food joints despite being incredibly wealthy and demanding her ransom on the city of New York be paid in quarters should send red flags. And no, unlike the above she never has mellowed (and never will).
Spider-Man himself is this. He acts as the silly one of every group he is in except with Johnny Storm ,who is as equally silly, and Deadpool, whom for obvious reasons is even sillier. However beware if you try to hurt his loved ones.
The Tick: And, isn't sanity really just a one-trick-pony anyway? I mean all you get is one trick, rational thinking. But when you're good and crazy, oooh, oooh, oooh, the sky's the limit!
The Séance from The Umbrella Academy. Although admittedly it's his constant use of drugs that cause him to be as crazy as he is, he still qualifies.
"Ever wonder what a vegetable thinks about? Firecrackers... bee stings... happy face eggs."
Hay Lin from W.I.T.C.H.: very quirky and artistic, cheerful and absent-minded, constantly writing on her palms and using her own expressions like "spacious" (instead of "cool") and "weirdific."
Marrow during her time with X-Force. She seemed to be completely dissociated from reality, barely able to focus on the task at hand, and spent most of her time babbling to an imaginary someone she addressed as "baby." This was later revealed to be her unborn child; she sacrificed the baby to regain her powers, and had suffered a psychotic break as a consequence.
The Flash: Bart Allen, especially as Impulse as he appeared in Young Justice and his self titled ongoing, is a perfect example of this trope. He was raised at Super Speed in virtual reality in the 30th Century, so it's kind of unavoidable.
X-23's younger clone sister Gabby aka Honey Badger tends to fall into this trope owing to her general innocence as well as having been raised in a lab. She herself admits to thinking a lot of weird things. Of course, this also allows her to become instant best friends with Deadpool.
In Mélusine, we have Mélusine's cousin, Mélisande. Combined with low intelligence, a distracted mind and Literal-Minded, Mélisande can't understand a simple greeting phrase and seems warped in her own universe. Seriously, she makes Elan look like a Nobel-prized genius.