"I'm going to work on a shrimp farm. Or something. Some kind of farm. Do they farm crabs? I mean, they must. It's not like crabs just walk into the grocery store. Though I guess they could if they wanted to. Wouldn't that be funny? I don't know, maybe not. But I guess if I was farming crabs I might get pinched. What?"
Dungeon Keeper Ami features Cerasse- one of the fairy sisters. She has a thing for Tserk, one of Ami's minions... A tentacle monster- to the extent that she has a great deal of trouble prepareing calimari ("Oh, I can't! The poor little tentacles!"). When the sisters contemplate 'rescueing' Jadeite from Mercury, she is quick to volunteer that they ought to rescue Tserk as well, earning her some strange looks...
In Teenage Jinchuriki Shinobi, Michelangelo is the only one not disturbed by Jiraya and Gai's poses and behaviorisms, something of which the latter was enough to make Naruto warn his team to not look.
In All the Dementors of Azkaban Harry is depicted as nearly as much of a Cloudcuckoolander as Luna, from trying to "make friends" with Dementors by offering them chocolate to painting Azkaban pink at Luna's request to cheerfully running errands in Diagon Alley for prisoners like the Lestranges.
In Session Transcripts brain damage from more-abusive-than-canon treatment by the Dursleys results in Harry being extremely Cloudcuckoolander-ish when he shows up at Hogwarts, spouting random remarks like "I can't eat a table, but I tried to eat my door once because I was so hungry. It didn't taste very good and I chipped a tooth." and "Plants talk to me sometimes, but sometimes they're allergic to me and I make them sneeze."
Ganmed Lomust in Hivefled. Spacey, Obliviously Evil subjugglator who writes truly terrible fanfiction. One of the more popular fantrolls, probably for this exact reason.
Harmony Theory: Blaze's speeches are enough to make some fans suspect him to be Pinkie's descendant.
A Brief History of Equestria mentions that most contemporary portrayals (mainly Hearth's Warming Eve pageants) of Chancellor Puddinghead are severely downgraded from the original, whose behavior bordered on The Caligula mixed with Groucho Marx. In other words, when Pinkie Pie was ranting about 'thinking inside the chimney' and wearing a map on her face, she was trying very hard to be solemn and dignified.
Mirror!Rarity in The Dark Side Of The Mirror Verse. Her misunderstanding of social rules and fashion ideas in general is so mixed up, she genuinely thinks her bizarre fashion tastes are completely normal and following the rules, and that her vandalism is simply improving things. Justified, as her parents stuffed her head so full of contradictory ideas and rules, she ended up not entirely sane trying to apply ALL of them at once instead of applying different ones to different situations. She gets a bit better thanks to Mirror Flim and Flam, but still remains a bit out there, such as trying to apply fashion to Twilight's Superpowered Evil Side.
Trying to be normal. He wondered what that was like; to try to be normal. Most people were normal, or normality came naturally. Or behaved however they saw as normal. Even most crazy people thought they were normal. Breech was trying to mirror their behaviour to be normal, and like a mirror, it came off wrong. Identical in every way, but somehow wrong; a facade, an illusion. What must it be like, to have to try to be normal?
Always Having Juice has the alternate universe version of Metal Sonic, whose mental faculties appear to be malfunctioning. He believes he is "The Real Sonic", and frequently tries to prove it using stripes, the true meaning of Hanukkah, and the letter F. The viewers have caught on to his nuttiness, however, thus resulting in questions such as "what is your favorite flavor of frisbee?" (for the record, it's Satanic Death Christ).
Oddly enough for a war story, Private Lenham Remer in The Legacy Of The Blood Ravens is a perfect example. When he's not kicking ass with his squad, he can be found musing about subjects like Space Bears, and if Space Marines can ride them.
The sad thing is, these two examples are him being coherent in the context of the scene.
More extreme examples of his Cloudcuckoolandiness include "I like pudding!" and "I farted, it's funny!"
Marik is this and Poke the Poodle. His evil plans include pushing Yugi off a boat so that his hair will get wet and it'll take hours to dry, and borrowing Yugi's puzzle and not giving it back to him...for at least another 2 months. He also invited Dan Green to the Evil Council because he played a villain in one of the Pokemon movies, and seems to believe that there is a difference between killing someone and just 'destroying them a little'. And then there's the standard weirdness.
Marik: By the way, Bakura, your costume could use some work. Bakura:... what? Marik: I mean, I don't like to criticize, but it looks like your mother made that for you! Bakura: Marik, this isn't a costume. It's my clothes. Marik: Well, you could have done a better job is all I'm saying! Bakura: A better job of WHAT?
Vegeta: Nappa! No! It's a trick! Nappa: But Vegeta! Trix are for kids...
Major Raikov in the Metal Gear Solid fanfic Stray is "always the type who took sanity more as a suggestion," to the extent that Ocelot considers Raikov's Psycho Electro boyfriend to be the stabilizing influence in the relationship.
raocow takes on a persona like this in his Let's Play entries. See: his Kamek's RevengeLet's Play, where he declares a fear of circles and refers to a flower that spits items when fed an egg or enemy to be "an altar to our dark flesh god".
He has weird names for the characters of Super Mario World. He calls Yoshi "stupid horse" even though he is a dinosaur. He calls the Charging Chuck "Football Charlie" because they are in football player uniforms.
To quote Raocow himself: "I'M SO GODDAMN WACKY"
Demykins from the Kingdom Hearts fanfic Those Lacking Spines behaves in this manner, with a multitude of his lines as shoutouts to other media; when he doesn't quote from various works, he ignores the situation altogether.
Which is also the common fanfiction depiction of regular old Demyx, because 90% of the people who write fanfiction have absolutely no concept of what makes Demyx Demyx. Or any other character, themselves.
Viole from the "Towerof God" fanfic Dei Fabula; About to get burned by flames that can melt rods of steel in milliseconds? Ice-cream daydreaming time!
Not to mention that prior to that, his method of getting rid of a parasite that was mind-controlling the neighbourhood's friendly firebreathing Damsel in Distress was to french kiss her as opposed to simply sticking his fingers down her throat like his canonical counterpart did.
Farfalla's note before one Fan Fic Header: I sometimes refer to slashy bits in canon as 'sardines', because they're thrown like little rewarding treats to us little loyal dolphin-like fans... but that has nothing to do with this story.
Enter Ken Finlayson: The titular character is viewed as one by the people of Norrisville due to his belief in the cryptozoological, the extraterrestrial and the supernatural but he really isn't. Still, many in-story believe him to be.
Eric the Hedgehog from Sonic X: Dark Chaos embodies this trope. He's an Idiot SavantTalkative Loon who loves breaking the fourth wall and talks to his weapons like they have personalities. Deconstructed later on - not only is it revealed that he's physically brain-damaged due to imperfections in Maledict's cloning process, but he's fully aware that his thoughts aren't entirely normal and has hidden crippling depression as a result.
Ignacio "Cio" Lumbre in Seven Devils is this trope, considering that when there was a fire, he tried to save the puppies before himself. Of course he did start the fire, and felt a little guilty.