By the end of Serial Experiments Lain, most of the characters, major and minor, including to some extent, Lain herself.
Hell, it's implied to happen all over the world.
Newscaster: Lets all love Lain! Lets all love Lain! Lets all love Lain LainLainLain!!!
Happens quite a few times in Code Geass, with results ranging from a Heroic BSOD to Insane Laugh... to attempted hallucinogenics use. And attempted murder-suicide, on a couple occasions.
Not to mention the final arcs of each season, though the second one is more ambiguous, depending on your opinion of Lelouch's logic (hint: either bringing the world to it's knees and united in mutual suffering then assassinating yourself is a good plan, or Lelouch is mentally unbalanced. Or both.), the first is clearly a plan to force someone to shoot you, then blaming them for it, with no obvious measures to limit the likelihood of dying. Not to mention Lelouch's attempt to trap himself and his father in a giant chamber, devoid of food, dooming them both to starvation, if not for the insufficient explosives.
One particularly notable incident from Code Geass is when Nina goes full on Yandere, to the point of showing up with a nuke to kill Zero, even though it would likely take everyone around with it, even her friends. She sounded beyond unhinged at the time, wild-eyed and shouting with grief and anger. Thankfully, it was defective...but then Nina switched to working on Applied Phlebotinumfantastic nukes. Oy.
Tengen Toppa Gurren Lagann has Simon going Ax-Crazy while fighting enemy mechas, locking himself on his room almost 24/7, and spending all his time carving statues of Kamina out of stone. Luckily, he finally does a 180° flip.
An entire episode of Elfen Lied is dedicated to the Ax-Crazy protagonist Lucy's childhood, in which we get to see just how she got so unstable and murderous through this trope.
In the 2003 Fullmetal Alchemist anime, Scar's older brother is revealed to have suffered this. The combination of his lover's death, failing to revive her through human transmutation, losing his genitals in said transmutation, and the imposing threat of a war on his home eventually pushed him over the edge. Using his own body, he planned to create a Philosopher's Stone out of Ishval's people during the massacre. He wandered out onto the battlefield completely naked with Ax-Crazy eyes and tears streaming down his face as Scar watched in horror. It didn't end well.
In Bleach, Ichigo exhibited some Sanity Slippage when his Super-Powered Evil Sidereally flipped out and completely stomped Ulquiorra. Uryu tried to stop Ichigo from further slashing his body; he received a sword to the stomach for his trouble.
In Romeo X Juliet, after witnessing his father Titus's death, Mercutio quickly starts losing it. And he ends up killing the murderer, Lord Montague, through quite the Karmic Death.
Professor Stein goes completely insane as a result of Medusa's plans. He got better, but still the entire collection of scenes showing his descent into madness is creepy.
Sort of applies to Soul Eater as well, once he started using the black blood.
The entire cast is constantly teetering on the edge of this trope. Madness functions as The Dark Side in the Soul Eater universe, with characters falling into madness becoming evil and others resisting the temptation to madness. One of the major villains is someone who slipped so far into insanity that he became an Eldritch Abomination.
The manga-verse now holds that he's apparently insane because he's an Eldritch Abomination personifying a kind of madness. Same point reveals that Death the Kid is the next in line for slipping.
In Gankutsuou, we get this when the Count of Monte Cristo watches Albert whom he had grown very fond of, leave his ship to witness the start of the Count's revenge on Albert's father. At first he sobs desperately and then he breaks out into hysterical laughter. And from then on, he's out there, madly pursuing his revenge. Andrea Cavalcanti also does this, going from creepy to outright deranged by the end of the series. And let's not forget Danglar's Death by Materialism, preceded by a good amount of madness...oh, and Fernand Mondego, who snaps to the point of shooting his wife and son. Or Héloise de Villefort. Yeah, this show is pretty much filled with the trope.
Gyagu Manga Biyori has the tale of the son and father who wanted to hire a tutor (and chose her more for the sake of her looks than the help she would give), but take many days to realize that they were tricked (Of course, like everything else, it is done in such a way that you have to laugh). At first they just look depressed when she doesn't show up, then their hands start bleeding because they clench their fists so hard and are beating the walls... their explanation for her absence is that they don't have a welcoming enough feeling in their home. Cut to them dressed as clowns as the father keeps stabbing a rock-hard cake while shouting "She won't come!" several times, mixed with laughter... It becomes an Inverted Trope when they then grab a hold of themselves and they work their way to becoming pure levitating Buddhists.
Itachi displayed this during his fight against Sasuke, while claiming that his true plan was to get Sasuke to awaken his Mangekyo Sharingan so he could use his eyes to replace his own, and suggests this is his true self. It actually isn't, as Sasuke later discovers.
Sasuke plays it straight, though. As if deciding to murder everyone in the Leaf Village weren't enough, he outright hits the floor in chapter 480 and the few after it. He's basically been reduced to a homicidal maniac who will kill anyone if they become the least bit inconvenient to him.
Orochimaru went through this in his backstory, but managed to conceal it enough that he'd already been kidnapping people for depraved experiments for ages when Sarutobi finally worked it out. Nagato went through something more like a Heroic BSOD that resulted in a Madness Makeover to Pain, but there was enough of a time spread that he definitely had time to devolve gradually. As Madness Makeovers go, wearing your murdered friend's corpse as a puppet avatar is pretty near the top.
Gaara qualifies for this from his flashbacks. As a kid he was initially sweet and misunderstood. Then his father started sending people he loved to assassinate him. Fast forward to his early teens, and he's a homicidal maniac talking to sand as though it's his dead mother. Luckily he recovered.
Although as it turns out he's pretty much right about the sand. His sand automatically shielding him is basically his mother's spirit moving to protect him from harm.
Kabuto lost quite a bit of sanity after Orochimaru died.
His backstory reveals that he suffered a significant dose of this when he was nearly killed by his mother figure, who in turn didn't recognize him. Increasingly unsure of his own identity after years as a spy, only Orochimaru's intervention stopped him from cracking.
In Stepping On Roses (Hadashi de Bara wo Fume), Nozomu, who at first seems like a really Nice Guy gradually becomes more and more of a psychotic Yandere as the series goes on.
In episode 7 of Puella Magi Madoka Magica, we can see Sayaka breaking down fast as she fights a witch. Why? Because a friend of hers had just told her that, in one day, she was going to confess to the boy Sayaka loved. That was just the last straw, though. Other factors certainly helped.
By the end of Rebellion, it's clear that Homura may very well have left her mind in that Flower Field. And that's not even getting into what Kyubey goes through.
In Gregory Horror Show, the Hell Hotel can only be found by people who have already begun to slip a little (the first guest may have been going mad with boredom from his monotonous life, and the second guest starts to slip when her best friend marries her ex-boyfriend). Prolonged exposure to it breaks the guests more and more. Both protagonists have completely lost their sanity by the end of the series.
Youji Kudou begins slowly losing his mind after killing his would-be love interest before she could kill him first (although close scrutiny will reveal a few hints of it beforehand, such as a manga scene where he mistakes Omi for her after waking up from a nightmare). Before it's over, he's nearly killed several one-night stands and had several long conversations with the woman's ghost.
Ken Hidaka doesn't fare much better: starting with the two-part OVA Verbrechen ~ Strafe, he finds himself enjoying the act of killing more and more, goaded along by having to fake killing his teammates and having to actuallykill his girlfriend when she turns into a target. He manages to avoid falling so far as to start killing people on his own time, but in the Radio DramaFight Fire With Fire, set after Weiss has been dissolved, he tries to convince Aya to become the new Persia so that he'll have someone to tell him who he can kill.
Gintama: Yamazaki gradually loses his mind as a result of having survive on nothing but milk and anpan over the course of a month-long stakeout. First, he starts sparking anpan into the faces of random people, then starts seeing and hearing nothing but anpan, then finally snaps, with one of his log entries consisting of nothing but "ANPANANPANANPANANPAN". He never completely recovered from the incident, with him having random relapses in his insane behavior from time to time.
Quite a few characters in Pandora Hearts go through this at one point or another, like Leo after Elliot's death or Oz when he goes into Love Makes You Crazy mode for Alice. (He gets slapped out of it by Gil).
Mirai Nikki: Yukiteru. Yuno doesn't count because that would imply that she was sane to begin with.
A bit in-continuity, but mostly during the SEEDs flashback, after Rem gets Vash and Knives out of the medbay where they learned the awful truth and promptly attempted to starve to death. Vash persists in wanting to die and hating humans, stabs Rem in the side in the course of a fight over a fruit knife he hopes to use on himself, and for a really awful moment, he smiles. Then he breaks down crying, patches her up, and things start to be okay again.
Notably, Knives didn't get this. Knives went promptly and utterly over to mad from the revelation, and acted like he'd blocked it out and was perfectly okay right up until enacting Plan Kill All Humans.
Legato also devolves steadily and horribly over the course of Maximum—yes, he was already bonkers, but after Knivesbreaks his spine he falls apart rapidly. Though he doesn't get less dangerous—after all People Puppets are his specialty.
Happens to Jeremy in A Cruel God Reigns. He could also have some possible Madness Mantra, at one point repeating "Hallelujah" over and over again while Greg ties him up between his bed posts in a very crucifix-type manner and whips/rapes him. Jeremy also laughs at the news of Greg's death (which he caused), tries to seduce his uncle, and plays mind games with Ian and generally seems unhinged the further the story goes.
In Attack on Titan, several characters experience varying levels of it due to the very nature of the world. Anyone that hasn't had a Freak Out, slowly gone mad, suffered a Heroic BSOD, or otherwise shown signs of coming apart at the seams probably died before they had a chance to crack. The trauma the survivors experience is a major theme, and teetering dangerously close to the edge is not only common, but completely understandable when the risk of being Eaten Alive or watching your friends suffer the same fate is an everyday thing.
Bertholdt seems to become more and more hysterical after The Reveal.
Reiner Braun, after The Reveal. The fact that he's been teetering on the edge of complete insanity for a long time is particularly shocking, with his Villainous BSOD continuing to get progressively worse.
In a side story, an Aberrant Titan is shown to be experiencing one. It goes from calm, to whimpering, and finally begins to cry while clawing at its own face until it nearly tears its own eyes out. Then it goes back to normal behavior, and eats the unfortunate soldier it had caught.
The mother and father of sanity slippage: Kars from JoJo's Bizarre AdventureBattle Tendency. After attaining Ultimate Life Form status, he loses anything resembling sense. And once he's thrown into space, he goes so insane his brain shuts down.
Liang Qi, not the biggest bastion of sanity to begin with in the Canaan series, rapidly starts slipping after Alphard, who she has an obsessive devotion for, abandons her, resulting in a Villainous Breakdown that ultimately leads to her having to be given a Mercy Kill after she gives herself a dose of the Ua virus in a mad bid for Alphard's attention, only to learn very painfully that she doesn't have the genes for synaesthesia.