His Let's Play of Eversion is his best example of pretending to lose his sanity due to the game's Nightmare Fuel. His blind run of Hell Temple from La-Mulana is his best example of really losing his sanity due to the game's unrelenting cheapness.
Alfred Alfer, the dog, has hit the event horizon of insanity by part one of Alfred's Playhouse due to the aftermath of loneliness and his childhood abuse and neglect, whereas in the Alfred Alfer Shorts he appeared relatively sane by appearance, aside from being a Cloudcuckoolander. He had a best friend, Phat Dawg, and held a job at a generic Taco Bell stand-in. Things had apparently turned sour in his psyche when, though a random occurrence, Alfred's memories of neglect and loneliness resurface as does his alter ego, dictator Pickles, in The Rise of Alfred Alfer. The Perpetual Limbo of the Room appears to be the segue from Alfred's initial psychotic break into the birth of the physical Playhouse.
Speaking of raocow, he goes in completely the opposite direction. Normally a Talkative Loon, he becomes very... sane and quiet when levels start to grind at him. Some Let's Plays have nothing but angry ranting.
The epic LP of Sonic the Hedgehog (2006) by pokecapn, medibot, Kung-Fu Jesus, IlluminatusVespucci and John Condit. They played the entire game in less than two days, playing almost non-stop, with only one of them sleeping at any point. Most involved are mildly Cloud Cuckoolander-ish to begin with, but as the game progresses, their commentary is filled more and more with non-sequiturs and Cluster F Bombs. A quarter of the way through, they've ordered and consumed copious amounts of General Tso's chicken because it's "full of ideas" (and thus, eating the chicken will give them ideas. It makes perfect sense!). Halfway through, one of them ironically blares out an a cappella version of Flashman's theme at random moments. Three quarters of the way through, they get stuck in the last level, and over the two hours it takes them to beat the level, they can be heard cycling through the Five Stages of Grief (denial, anger, bargaining, depression, acceptance). By the end, one of them is reciting song lyrics in Shatner-esque spoken-word renditions, including "Rocketman", "Fast Car", "Down Under", and finally "Thriller" over the final boss fight. When combined with the slow, melancholic choral music used in the later stages, it borders on being strangely touching, depending on your mood.
It didn't help any that, besides Medibot, none of them got any sleep. Even Medibot didn't get much; the portions of the LP where he's missing don't add up to much more than a catnap or two (and he apparently went home for those, cutting the sleep time further).
On a similar note, the Freelance Astronauts attempt at beating The Legend of Zelda: Ocarina of Time (master quest version) descends into this between the forest and fire temple. Ferr begins making jokes about the gilded age and Maxwell gets the group stuck inside of a fire maze. The group had also been attempting a "Fantasy Race" with the Lord of the Rings trilogy, further compounding the insanity.
During LittleKuriboh's LP of Duke Nukem Forever, he gets stuck for one hour on a boss battle over the Hoover Dam. As time and deaths pile on, LK talks about random subjects, goes through many of his character voices, jumbles words together and repeats them ad nauseum. The last straw happens in Part 26 (at 13:00), when he's unable to complete a quick time event needed to kill the boss despite tapping frantically on the buttons (turns out he was tapping the buttons too fast for the game to notice). This makes him laugh maniacally.
LK: [QTE fails and he's thrown to the ground] Are you kid— Are you— ... (laughter) Why does— Why— Why does— things not— Why— Why do things not do thing why are things in the... things?!
He then spends most of Part 27 babbling incoherently about mushrooms, badgers, and barrels.
LK: I may have lost my sanity... thank you game... for taking sanity...
Duke: Killing you is as easy as breathing.
LK: I have five rocket... Did he say killing you is as easy as breathing? What— it's not! If this is how difficult breathing is BREATHING IS THE HARDEST THING IN THE ENTIRE WORLD! ... Mushroom style... I'm doing this Mushroom Style. Mushroom style all over the place. Mushroom. Badgerbadgerbadgerbadger. Gimme the mushroom!
Marble Hornets is essentially made of this and nightmare fuel. Any character that is in front of the screen for several episodes is shown to become either paranoid to the point of obsession (Alex and Jay) or homicidally insane (Masky).
In her stay at the hotel, Jessica also appears to succumb to this when she realises that something is wrong.
As events progress and he finds himself in increasingly dangerous situations, Gordon in Freemans Mind is showing increasingly common bouts of uncontrolled rage and paranoia, which is impressive considering his original personality. This was perhaps most clearly demonstrated in episode 29, when he threw a grenade at a large pile of explosive crates.
Gordon: I HAVE TO BLOW EVERYTHING UP! IT'S THE ONLY WAY TO PROVE I'M NOT CRAZY!
Kevin Low from New Game Plus The Rpg, after countless slights against him, begins to believe the world is against him, as he begins to slip even more, he starts to have spans of time where he wishes vengeance upon his friends. "He smiles at the twitching bloodied body, he knows she is still alive, but he wants her to suffer. She has become a proxy for every single person who hurt him, every situation bent on beating him down. and now they all are suffering. and it pleases him."
The Nostalgia Critic went through one during his review of The Neverending Story 3. He started the review feeling scared and hateful towards the film as he does towards most films he thinks that are horrible, then in the middle of his review he goes off on an unusually long (for his standards) Cluster F-Bomb rant over the Character Derailment of the Rock Biter, and by the end of his review he insanely laughs while leaving his house to buy a crowbar, goes on a primitive rampage on the DVD with said crowbar, and tops it off with fucking the DVD.
Played for Drama in To Boldly Flee, as his guilt over losing Ma-Ti gets his suicidal tendencies into gear and makes him think that even Internet laws are all his fault.
This LP of Super Meat Boy. Hamster seems relatively okay most of the time until he starts dying a lot. Then he devolves into swearing, flailing, laughter that really just says "kill me now" and incoherent gibberish.
Mr. Anime, he was originally just a guy who reviewed anime until he went crazy and went on a killing spree killing his family and his teacher.
Note this is not a joke or an in-universe thing. He actually did so. Worst part is that a month before he did so her released a video that said that he was planning something big.
Lanipator's let's play of Amnesia Justine. He plays through the entire game an goes for a 100% completion to get the Easter Egg. After many, many deaths (you have to start from the beginning if you die), he gets to the end of the game and he doesn't seem to get the Easter Egg. He did get it, but he didn't know it. He later expressed this revelation with a comment which was 90% F-bomb. After that, he starts over, kills all the prisoners laughing like a madman and yelling at the characters in the game, and stands next to the last prisoner so that they could die together. Then, when the screen goes black, the words rebel for life appear.
The Nostalgia Chick snapped completely around "Sleepless In Seattle vs. When Harry Met Sally", kidnapping Todd in the Shadows, tying him to a chair and her twisted mind calling it a date. You wouldn't think it was possible, but she's only got worse from there.
Ever since he moved from his bookcase to a bar, Ask That Guy with the Glasses's sanity has been falling apart. Of course he was never sane in the first place, but it's almost sad to see him go from a Wicked Cultured chessmaster to a Jaded Washout who breaks into tears randomly. (So essentially he's now an evil version of Critic.)
This happens to a book in The Wanderer's Library story Of The Metabible. The Metabible, a godlike entity over which wars have been fought, is an impossibly long book. As you read farther it begins to get more and more incomprehensible, and by the millionth page it's completely broken and begging for somebody to read it. The last page of the book separates itself from the Metabible and forms a second one. It's implied it eventually goes through the same process.
At the climax of WormTaylor, already a mess of neuroses, suffers irrevocable brain damage from Panacea altering her powers and begins rapidly losing her faculties, starting with the ability to comprehend language, and eventually culminating in a complete Loss of Identity.
Arguably, Harrison Laine on season 2 of The Music Video Show. He wasn't completely sane in season 1 but, after the incident during the end of its finale, something isn't quite right with him...his love interest is a Fluttershy plushie. Also, there was that thing that happened in episode 32... Oh, and then there's episode 49...holy fuck.
In Ten Little Roosters, the Rooster Teeth crew is slowly suffering from this as they're trapped in their studios with everyone knowing they killed someone and there's a killer among them. Chris was already somewhat like this, running around dressed up like a Hobbit from Lord of the Rings and later learning that he had a mess of problems, but the others start succumbing to this as time goes on. As of episode 4, Ryan is showing the worst of it, haunted by the deaths of Gavin and Michael and fearing that he's developed a split personality. Ultimately, Ryan's incidents were revealed as a ruse by the real killer, Barbara, in an attempt to convince Ryan really was going insane and blame him for the murders.