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!!!
The eponymous Lenore 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.
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.
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.
"... 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?"
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."
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.
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.
The Adventures of Tintin: Professor Calculus in 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. Any of the Calculus scenes in this particular book could be considered a Crowning Moment of Funny.
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.
Bart Allen, especially as Impulse as he appeared in Young Justice, is a perfect example of this trope. He was raised at superspeed in virtual reality in the 30th Century, so it's kind of unavoidable.
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!
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.
The Séance from The Umbrella Academy. Although admitably 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."
Essentially all the kids from One Big Happy qualify. Central character Ruthie is pretty obvious, but she has even weirder friends, like vacuum cleaner fetishist Earl.
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.
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.
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.
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!?
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 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!
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 appearences 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).
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.
Ms. Wormwood: Calvin, pay attention!! Now, what state do you live in? Calvin: Denial. Ms. Wormwood:(sighing) Well, I suppose I can't argue with that...
—>Calvin (during the pledge of allegiance on the first day of school): I pledge allegiance to Queen Fragg and her mighty state of Hysteria.
The very fact that he sees Hobbes as real and everyone else doesn't (which often makes them suspect that Calvin is insane) also establishes him a place in this trope. The very fact that Hobbes thinks Calvin is crazy would fit him into this trope if that didn't. There's also the time where he comes into his class dressed as his superhero alter-ego, Stupendous Man. His classmates' facial expressions are the natural reaction anyone would have toward a Cloudcuckoolander.
Jon Arbuckle from Garfield, since the late nineties, has gone from a slightly-dim, arrogant loser to a full-fledged Cloudcuckoolander in some strips, with lines of pure insanity like "I think my feet are jealous of my hands because they get to point at things." This without even getting into the surrealistic brilliance of Garfield Minus Garfield and other projects to improve the strip.
Garfield himself had his moments in the strips (before he got Flanderized into a full-time Deadpan Snarker). Remember when he became Banana Man? Or Amoeba Man?
The animated special Garfield's Feline Fantasies was all about his mind wandering off into fantastic stories.
Meanwhile, US Acres had Bo Sheep, who showed more tendencies of this in the comics than the Garfield and Friends animated series.
Krazy Kat. S/he thinks that getting bricks thrown at her/his head is a sign of affection.
Get Fuzzy: Both Bucky and Satchel occasionally display these tendencies, and many of Bucky's feline visitors really do.
The entire Dick Tracy comic, since Max Allan Collins left. Bad guys getting squashed by steamrollers or having their eyes gouged out; businessmen (both good and bad) who dress like playing cards; characters being incinerated in giant fireballs; hillbillies defending themselves with bear traps; and every once in a while, something that seems to make sense. Rarely.
Peppermint Patty. To name but one example of her weirdness, she thought that Snoopy was an odd-looking little kid for years.
Hillary's classmate Nona from Sally Forth. Tends to take Hillary and Faye's idle Zany Schemeing and run with the idea into surrealism. All in the same tone of voice one would normally use when discussing lunch.
A few characters from the Ink Pen comic strip is this, the main example being Captain Victorious. Here, for instance. Or here.
Transformers: Wings of Honor: Vortex and Metalhawk get this way when sprayed with chemicals which mess with their processors. Sprocket is revealed to be this way after some feedback when he integrated into a ship. Now he believes that non-sentient machines talk to him, and speaks to imaginary friends. It's gotten to the point where in the final issue, he doesn't even know his brother died, and continues to talk to him as if he was still there.
Pierce from Zits seems to be always doing his own thing, on impulse, with little regard for consequences.