The Christmas episode of Adventure Time shows that, centuries ago, before The End of the World as We Know It, this happened to an antique collector by the name of Simon Petrikov after he made the mistake of wearing a cursed artifact. A thousand years of sanity slippage later, the man he was is completely gone, and what remains only knows himself as The Ice King.
It gets worse. Later episodes document Simon's attempts at surviving and raising Marceline after the nuclear holocaust, and his desperation at remaining himself.
"But I'm losing myself and I'm afraid you're going to lose me too."
Being harassed 24/7 by the spirits in his castle (that only he can see, thanks to his wizard eyes) probably hasn't helped any, either.
Stan: Oh am I!? Would an insane man try to drink you?! (holds up Klaus' bowl to his lips and starts drinking the water)
Francine undergoes some in "Stan of Arabia" after being forced to become a subservient wife under Arabic laws, culminating in her breaking into a musical number about how Saudi Arabia is the worst place in the world while dancing nearly nude and breaking as many laws, norms and taboos as she can.
Aang went through one half a season previously due to sleep deprivation. He started out with just encouraging everyone to prepare for alarmingly specific things to go wrong and ended up having a several-minute hallucination of a swordfight between Momo and Appa while a six-armed Guru Pathik floated past singing about chakras.
While the Red Guy from Cow and Chicken isn't exactly sane to begin with, he ends up having a complete mental breakdown in the episode "I Scream, Man" as a result of taking a job that he despises: as an ice cream man. Predictably, he ends up in a mental clinic.
Red: AAAAAHHHH!!!!! I CAN'T TAKE IT ANYMORE!!!! I don't have any ice cream! Not one, so! EVER! (Hangs near the side of the cop, then switches to a calm, quiet voice) I don't even like ice cream, (switches back to Large Ham mode) OR KIDS! It says "Eye Screem" on my truck, not "Ice Cream"! Get it? 'cause I SCREAM!!! Look at me. (screams while spinning in circles) SCREAMING IS MY HOBBYYYYYY!!!!!!.
Policeman: (hits Red with a sausage) It's the padded cell for you, scream man!
Eddy loses it on a few occasions, such as in "Laugh Ed Laugh" when all the Kids get chicken pox, leaving him with no one to scam.
"Little Ed Blue" has Ed become massively pissed off for apparently no reason. He had a pebble in his shoe. This revelation drove Eddy crazy too. But when Eddy slaps him in an attempt to snap him out of it, Ed goes into a unstoppable rampage, eating metal slides and nearly snapping Plank in half.
Kevin starts going manic from paranioa when the Eds go missing in "No See Ed."
Edd goes insane in "Cleanliness Is Next to Ed-ness" from not being able to take a shower after his parents begin renovating theirs.
Futurama: Fry suffers this in "Insane in the Mainframe" as a result of being a human confined to a robot asylum.
In "Mobius Dick", Leela went insane when she became obsessed with trying to kill a space whale as revenge.
Amy: You've gone from crazy like a fox to crazy like Fox News.
If you look closely at some of the text in Journal 3 on Gravity Falls, it seems like the Author underwent something like this. He did indeed, as a result of being manipulated by Bill Cipher into building a device that would destroy the world. However, his case was relatively mild and he managed to pull himself back together - probably a result of his trip to another dimension.
Mabel slowly starts losing it in the season 1 episode "The Time Traveler's Pig" after Dipper's time travel antics cause her to lose her pet pig. After handing out calculators to people across eight centuries, she eventually breaks completely and starts banging her head against a post and doesn't stop for a month, during which time she grows vines and gets a shoulder snail. Fortunately, it's temporary, and getting her pig back seems to resolve all the problems, but this example is played more for drama than laughs, despite its absurdity.
Old Man McGucket helped build the machine in Stan's basement thirty years ago, but an accident with the device heavily traumatized him. He then built a gun that would erase the memory from his mind, creating the Society of the Blind Eye so that others wouldn't be freaked out or traumatized by the weirdness of Gravity Falls. However, repeated use of the memory wipe took a toll on McGucket's sanity, until he was reduced to the junkyard dwelling town kook he is today.
Dib of Invader Zim isn't that stable to begin with (being one of the handful of reasonably intelligent people in a Crapsack Worldfull of idiots will do that), but over the course of the series becomes increasingly desperate, paranoid, and manic, and talks to himself with greater frequency.
Played for Laughs on Jimmy Two-Shoes, when Jimmy is left the only person in Miseryville awake while everyone else is in hibernation. He slowly begins to go crazy.
The eponymous character of Kaeloo has lost her sanity several times, most notably in Episode 100, but because of Negative Continuity (except in the aforementioned episode) she is back to normal by the next episode.
Hank's barber Jack in King of the Hill started going senile, culminating in Bill mentioning something about Jack getting in a high speed police chase. Earlier we'd seen him saying he couldn't give Hank a haircut because he didn't have a stapler, spraying Hank's hair with peroxide before walking out of his shop and assaulting someone for their bicycle, and finally dumping shaving cream on his head and removing it with a hair dryer.
This is oddly reversed a few seasons later when Luanne starts working at Jack's shop and he's got all his cognitive abilities working in okay condition, implying the above was either a nervous breakdown, he'd been put on very strong medication, or bad writing.
In the episode "Pretty, Pretty Dresses" Bill is very lonely and depressed, and finally loses it when Hank destroys the gifts he'd been saving for his ex-wife Lenore in case she ever came back. He starts wearing her old clothes she'd left behind and talks in a falsetto voice, believing that he himself is Lenore; eventually Hank is able to break him out of it and gets him to get over Lenore.
In The Lion King 1 1/2, during a scene approximately taking place during Scar's Villain Song in the original movie, Timon is on the verge of cracking in his "dream home" quest...
Timon:(comes across a steam vent) Hey! This is home sweet home, baby! Home— (gets sent into the air a bit by the eruption) Ha ha! Steam! Ha ha! Steam is good. Steam is—is—is water. Whoo! Gotta have water. You know, for the dream home. Steam home, dream home! Steam, steam, steam.
Looney Tunes: Daffy Duck seems to fall victim to this a lot. A perfect example occurs in Duck! Rabbit! Duck! after he gets shot by Elmer Fudd one too many times:
Daffy:(hysterical) Shoot me again! I enjoy it! I love the smell of burnt feathers! And gunpowder! And cordite! I'm an elk — shoot me! Go on, it's elk season! I'm a fiddler crab — why don't you shoot me?! It's fiddler crab season!
Happy Rabbit (the beta Bugs Bunny) in "Hare-Um Scare-Um" does a song that tells how watching Looney Tunes cartoons drove him insane. Imitating a police officer, his throwing the book at John Sourpuss' dog tells all.
You know what this'll cost you? Thirty days...hath September, April, June and Montana! All the rest have cold weather! Except in the summer, which isn't often!
A year earlier, Daffy Duck himself tells how he did quite well working on a merry-go-round until it broke down.
My Little Pony: Friendship Is Magic: The Fan Nickname for this is "Cutie Mark Failure Insanity Syndrome", given that a pony failing to live up to their special talent (symbolized by their cutie mark) is the most common cause of ponies going nuts. (It's worth pointing out that in all of these cases, the insanity was cleared up by the end of the episode with The Power of Friendship.)
In "Party of One", Pinkie Pie thinks her friends are avoiding her because they're sick of her parties. Her normally curly hair goes straight, she makes her own party guests out of things like a sack of flour and a pile of rocks (whom she holds conversations with), her colourful persona becomes noticeably darker, as if it's under a shadow, and the normally insufferably cheerful pony becomes cranky, miserable, and borderline paranoid schizophrenic, all the while sporting a Slasher Smile. Her slippage even comes complete with Creepy CircusMusic (which has a disturbing resemblance to the leitmotif of a certain personification of chaos).
Twilight goes crazy again in "It's About Time" due to a warning from her future self that got cut off before the important bit could be delivered, leading to a spiral down into the depths of paranoia.
Princess Luna experiences this a thousand years before the time of the series. Literally consumed with jealousy and anger towards the ponies rejecting her beautiful night in favour of Celestia's day, she transforms intoNightmare Moon and tries to enact eternal night in a stubborn attempt to force the ponies to love her, showing that even a Physical Goddess isn't immune to CMFIS. In "Luna Eclipsed" she runs the risk of having to bear these insecurities all over again when the ponies begin to reject her once more, but fortunately Twilight is there to help.
Before "Games Ponies Play" aired, people were expecting another "Twilight goes crazy" episode. They were likely surprised when this trope was averted for that episode because Cadance taught her a controlled breathing excercise to relieve stress.
But if her hyperventilating near the climax of the episode is anything to judge by, it was a pretty near miss.
Rainbow Dash's turn comes in Tanks For The Memories, where her anxiety at having her beloved tortoise hibernate for the winter drives her to try and stop winter from happening.
The episode A Royal Problem shows us that not even Celestia herself is immune to this. After Starlight switches her magic with Luna's Celestia takes over her sister's nightly duty and expected them to be easier, only to quickly find out that because it's nighttime all the ponies in Equestria are asleep, which leaves her without company. She quickly start talking to herself. A lot. Luna admits to talking to herself as well.
Candace from Phineas and Ferb. It's not gradual and it's not regular like most examples, but it's definitely there. Seriously, this girl needs help. She has a psychological need to bust, it's shown at least twice that she can't go without it. That doesn't smack of good mental health...
If it wasn't clear before, the episode "Monster From ID", which takes place largely in Candace's subconsciousness, seals it. Candace's head is one messed up place.
Pinky and the Brain: Snowball in "Welcome to the Jungle", where he's the leader of a makeshift jungle tribe made up of tourists and it's implied he's been hallucinating.
Bubbles has one in "Bubblevicious" as she tries to prove to everyone that she is "hard-core".
Bubbles suffers this again in "Octi Gone" when she believes that somebody at the party she's at stole Octi.
Buttercup slowly goes insane in "Moral Decay" while stealing villains' teeth for money.
Whimsical Willy in "Neighbor Hood".
Mayor in "Hot Air Buffoon" when trying to save the town goes to his head.
Dick Hardly from "Knock It Off" becomes gradually more unhinged as his fortunes (and greed) grow.
Mojo Jojo in "You Snooze You Lose" totally flips out when he sees the Amoeba Boys have somehow built and assembled the device he designed to get rid of the Powerpuff Girls once and for all, and accidentally tricked the girls into getting into it.
Bubbles has one in the episode "Bubbles of the Opera" due to thinking she's losing her cuteness.
Buttercup loses it in "Clawdad" while on a boat to catch a crawdad, in the time span of two hours.
In Recess, we have poor T.J. in "The Box". Mrs. Finster introduces a new form of discipline in the form of "the box", a white box on the pavement, where troublesome kids have to stand inside as punishment. When T.J. is sent to it, he quickly breaks and is eventually reduced to a blubbering mess who rocks back and forth singing "This Old Man", constantly does stuff for Mrs. Finster while chattering about how he's a "good boy", screams at the sight of anything square-shaped, and thinks the walls are closing in on him while in the box. Fortunately, he gets better.
Rigby of Regular Show goes cuckoo when he learns the "Death Punch of Death", to the point where he's willing to drown in lava if it means beating Mordecai at Punchie's.
Ren of The Ren & Stimpy Show frequently falls into this. Stimpy sometimes does too. Their breakdowns range from hilarious to terrifying.
Ren: It's all so clear to me now! I've got it all figured out! I'm the keeper of the cheese! And you're the lemon merchant! And heknows it! That's why he's trying to kill us! We gotta get out of here, before he sets loose the marmosets on us! Don't worry, little lady, I'll save ya!
Rugrats: From the movie, Tommy totally snaps and almost pours mashed bananas on Dil, knowing the monkeys will be attracted to it. It's thundering and raining the whole time, and Tommy just starts acting more and more crazy as he starts ranting about how much Dil has ruined his life.
Stu also loses it when he's forced to take care of Angelica after she pretends to have broken her leg.
Fifty years of failure took a toll on Samurai Jack and his sanity as of Season 5; he gets frequent hallucinations of Aku's victims accusing him of having given up, and often argues with his subconscious as it tempts him towards less than moral actions (including once suggesting seppuku as a means of escaping his torment). Outside observers such as Scaramouch or Ashi just see Jack arguing and pleading with empty space/rocks, concluding that the samurai has well and truly lost it.
Homer himself does so in The Shinning segment of "Treehouse of Horror V" when the cable stops working and they run out of beer, all thanks to Mr. Burns.
Homer: So what do you think, Marge? All I need is a title. I was thinking along the lines of "No TV and no beer make Homer something something"... Marge: "Go crazy?" Homer: Don't mind if I do!! (proceeds to go crazy)
In "$pringfield", Mr. Burns' sanity erodes the longer his casino stays in business, Ala Howard Hughes. He grows a shaggy beard, becomes a hypochondriac, and when things fall apart, expects him and Smithers to fly home in a tiny model plane.
SpongeBob SquarePants and Mr. Krabs try to keep their sanity when they believed they killed the health inspector. Turns out that the inspector is still alive, and the Krusty Krab passes inspection.
In "Pickles", SpongeBob's life takes a turn for the chaotic after a Jerk Ass customer claims that he forgot the titular condiment. By which we mean he tries to cook his bicycle and hangs the toaster on his front door.
Poor Squidward. In between being the entire ocean's Butt-Monkey and victim to the constant shenanigans of his next door neighbors, SpongeBob and Patrick, it's no wonder he's been shown to break down into a sobbing wreck or fits of maniacal laughter when the Amusing Injuries and ever-present failures prove too much to handle. In "Squid's Day Off" Squidward tricks SpongeBob into running the Krusty Krab all by himself while he takes a day off, but he keeps imagining SpongeBob destroying the Krusty Krab and repeatedly goes back and forth to and from his house. While taking a bath he hallucinates SpongeBob is watching him and runs off to the Krusty Krab wearing only bubbles.
SpongeBob frequently has Sanity Slippage to the point that you wonder whether he's exactly sane to begin with...
(from "Stuck in the Wringer") Patrick: Hey, whatcha watchin'? SpongeBob:(watching static on TV) My favorite show. (the TV dies) Ha, ha, I love that part!
Mr. Krabs goes crazy when the squeaky noise of the rubber boots he gave Spongebob becomes too much to bear.
When Spongebob decides to live with the jellyfish in "Nature Pants," Patrick, who wasn't all therein the first place, goes ballistic, planning on catching Spongebob and keeping him in a jar.
On Stanley, Elsie experiences one when she grows tired of waiting for the butterfly to emerge from its chrysalis in "Caterpillar Countdown." The longer the wait, the worse she gets, as she really wants to see that butterfly.
In the non-canon Uncle Grandpa crossover episode, Pearl, who is the logical (most of the time), brainy perfectionist of the team, becomes increasingly unhinged and suffers at least three mental breakdowns from dealing with Uncle Grandpa's weirdness. This is even emphasized by a significant downgrade in her appearance and animation and major scenery chewing by her voice actress. It overlaps with temporary Flanderization because while Pearl is susceptible to bouts of hysterics and irrational behavior, she's never lost it to this degree before or since.
In "Chille Tid", it is implied to have happened to Jasper after she is trapped in Malachite by Lapis, when she sees Steven again, crawling towards Steven with an insane look on her face and a tone in which she sounds like she plans on eating him alive.
Peridot's stay on Earth has...not been good for her personal stability. Oddly enough, while in most cases of this trope there's a direct correlation between how conclusively one has cracked up and how much of a Jerkass one is, in the later stages of her SAN meter's decline, Peridot actually gets nicer, to the point where she's now officially a Crystal Gem; this realization led to her most impressive freakout to date.
Peridot: (into her tape recorder) I'm a traitorous clod! They'll never want to think about what I've done again!
(sighs, rewinds recording)
Peridot (recording): I'm a traitorous clod! I never want to think about what I've done again!
Peridot: (grins widely) YEAHAHAHAHA!
Steven: Peridot, are you gonna be okay?
Peridot (recording): I'm a traitorous clod! (rewind) -traitorous clod!
Thomas the Tank Engine: The Fat Controller, of all people, suffers this when he and Skiff are stranded "Skiff and the Mermaid". He becomes obsessed with survival on their small island - even though Skiff points out they're not far from port and can sail back once the tide changes.
Transformers Prime showcases Megatron's slow descent into (further) homicidal madness. Doping using the blood of Unicron certainly hasn't helped, but he's clearly slipping even without it. It becomes most clear when he get the Dark Star Saber, whereupon he starts shouting about how he'll tear this world apart (despite the fact that doing that dooms the entire universe).
Rabbit has experienced this at various times in the Disney Winnie-the-Pooh franchise, often having to do with Tigger. His thinking her has seven years bad luck because of a broken mirror in "Luck Amuck" from The New Adventures of Winnie the Pooh is just one of many, many examples.