Anyone that breathes in the Giygas gas in Attack Of Giygas. As for named characters, we have Dr. Mario, Pichu, Ashley (who claims that she's sane), Lucas, and Link.
Batman Beyond Revisited: The appropriately named OC, Chainsaw, a leader of a Jokerz gang that thrives on torture and murder to the point that the Joker himself would be disturbed.
In Supergirl fic Hellsister Trilogy, Satan Girl is a Supergir's duplicate, the embodiment of Kara's dark side with no conscience or restrain to control her murderous impulses.
Kryptonians could survive in space without a suit. Was that not a pleasure? It certainly was. She could live her life between the stars, and never once need to breathe. She could devastate planets, wipe them clean of life. Rebuild them at her whim. She could tyrannize worlds, whole systems of planets, make them bow to her mighty hand, instantly execute anyone who dared protest—or just anybody she wanted to kill. She could explore pleasures of the body that Kara never would have dared to, satisfy lusts that the blonde beast never even knew she had. She could force herself upon any suitor, male or female or whatever, and destroy them after their job was done. Or perhaps just maim them, so that they could never again do such a job for anyone else. Satan Girl smiled. Now that was being imaginative... She could have children from those couplings, or kill them in the womb. She could become a goddess to an unsophisticated planet's people. Drinking in their worship, demanding sacrifice. All of this she could do, she would do, and more. For Kryptonians and Daxamites were gods, off their homeworlds. They really were. What a pity their morality forced them not to realize that fact. She clasped her bent knees to her chest and thought. The problem was, in this time, she was hardly unique. Billions of Kryptonians existed on Rokyn. Billions more Daxamites, with the same power, existed on Daxam. Luckily, there was only one prisoner still left in the Phantom Zone, that old poop Gazor, so there wasn't much competition there. But, somehow, she'd have to do something about both planets. Daxam would be easy. A shower of leaden hail across its surface, and the dead would litter the ground in heaps beyond Hitler's and Stalin's dreams. That world would stink of corpses for eons to come. She laughed soundlessly.
Tyrian Callows is a deranged Serial Killer who is known for dismembering bodies. He was picked up by Torchwick to aid in harvesting bodies for his organ trade. Hazel says that being able to kill will be enough payment for him and he needs to be told to only slit throats as to not damage vital organs. He unfortunately can't keep his impulses under control when he's not on the job since he was responsible for a number of dismemberment that Blake and Sun were investigating.
The mob boss only known as "Cinder" has better control over her impulses compared to the two examples above. However, the fact that she gets sexually aroused by violence does not reflect mental soundness. She may be pragmatic, but she's definitely eager to commit violence when she can get away with it. Whenever she lets herself cut loose, she shows the trope in full.
Daemon from the Tamers Forever Series could be the posterboy for this trope, a nigh unstoppable psychopath who is fuelled by his own hatred, often exploding into violent rages and laughing maniacally at the sight of others in pain.
Joe Dark from Clash of the Elements is a very prime example of this trope and this very trait of his is amplified immensely after he gains the Dark King's powers.
Many villains, both Original and canon, in Ruby and Nora count, but none more than this story's version of Neo. She's a sadistic maniac who's introduction includes a level of violence that can only be described as absolute Gorn. When she hijacks an Atlesian airship, she has her fun blowing ships out of the sky.
Captain June Harper, Villain Protagonist of the Doctor Who story Gemini is a Serial-Killer Killer on the run from the Time Agency that she used to work for. She maims, tortures, and murders anybody she that deems to be threatening the innocent, and she takes great pleasure in the pain and fear that she inflicts on her victims.
Similarly, in Inner Demons, after Twilight is consumed by her Queen persona, she becomes an extremely mentally unstable sociopath who switches between cold emotional detachment and blind fury at the drop of a hat. It gets to the point where she even decides to kill her brother when he rejects her advances.
And that's before her Villainous Breakdown. Afterwards, she's left constantly switching between laughing maniacally and sobbing (it's actually implied that's her true personality trying to resurface and being forced back down by the Queen), and on top of that she's decided that rather than ruling Equestria, she'll destroy it.
Nightmare Phobia, the Big Bad of the Chaos Verse (an official spinoff of the POV Series above) is this trope in spades. And it's not helped at all by the fact that the more powerful she becomes, the more her memory is corrupted, which in turn only fuels her insanity more.
A Growing Affection has the self-named Reaper of Shinobi. She thinks that anyone who uses chakra (herself included) is a demon, so her goal is to kill all ninjas and then kill herself. Over the years she has come to believe the backstory she fabricated for herself (that she is the sole survivor of the destruction of her village in the last war).
Several characters in You Got HaruhiRolled!. The fic follows canon in portraying Ryouko in this way, but in addition, Haruhi is occasionally Flanderized into this trope, such as when she threatens to kill Kyon's sister for dishonoring the SOS Brigade. Additionally, Sasaki usually functions as the Only Sane Woman of her group, but in the chapter where she finds a Death Note, she arguably out-crazies YagamiLight himself.note Though this is only implied, rather than outright stated.
Twilight Sparkle is the prime example in Pages Of Harmony, though Rainbow Dash is definitely no slouch in that regard either. Twilight basically tortures and Mind Rapes her friends to extract their Elements, becoming increasingly deranged the further she goes down the road. Rainbow Dash, the first killed as a result of Twilight's actions, has her memory twisted and warped to the point where she cannibalizes herself as a result of no longer having Loyalty to even herself, and swears that she'll find a way to kill Twilight for what she did.
In Sinful Behaviors , Rudy becomes this when he is taken over by the black chalk.
Utakata, a.k.a. Saiken the Slug, in Kitsune no Ken: Fist of the Fox, is this. In fact, before he underwent the training that gave birth to the Kyuushingai, he was a homeless man with serious psychotic issues, but he had the presence of mind to hide that fact from the doctors and scientists who worked on him and the other eight members of the group. The end result...
Utakata: They allowed a psychopath like me to be transformed into a SMART psychopath.
In Mega Man: Defender of the Human Race, Dr. Wily and several of his Robot Masters are this. Both Wily and Bass love the sound of screaming, and Splash Woman goes berserk after Roll attacks her and threatens to destroy the entire cavern to get rid of her enemies.
In Sunset of Time, the Big Bad, Vesper Radiance. She's an insane psychopath who has personally murdered several characters already, attempted to kill even more, and her victims include her own parents out of revenge for putting her in an orphanage as a foal.
Tsukune's ghoul form counts to a T. In Act III chapter 52, when unleashed, he proceeds to brutally dismember and disembowel Kiria with his bare hands, laughingwhile he does it.
Jovian and Jacqueline Kikion also qualify, particularly the originals who served Hokuto. In Act III chapter 52, when Hokuto warns Felucia that Jovian will kill Ruby and Tsukune if Felucia won't hold them back, Jovian gleefully cheers that she loves killing. The resummoned Jovian and Jacqueline in Act VI are considerably more mellow.
Arial Kuyumaya often comes across as this in Act VI, in part due to her Hair-Trigger Temper and obsession with Dark; in chapter 7, the very first thing she does as soon as she discovers that Mizore is Dark's girlfriend is use her ice claws to literally tear Mizore apart in a jealous rage right in front of everybody; by the end, Mizore is so horrifically injured that Ceal has to transfer organic matter from deceased HDA officers to actually heal her. The others even outright describe her as insane on more than one occasion.
Even without chrono dementia, Evil!Falla is a sadistic, sociopathic bitch who can slaughter people by the handful without hesitation, while laughing about it.
As revealed in Act VI chapter 52, the Fairy Tale-created clone of Moka is little more than a savage animal.
In Prison Island Break, Shadow is a completely unpredictable psychopath. It's for a good reason though; he was locked in the Hole with Scourge after attempting to escape, Scourge basically being Ax Crazy and Nightmare Fuel given physical form.
Hope for the Heartless includes as minor antagonists a pack of more than thirty Savage Wolves, fittingly called the Mad Pack. They have terrorized the wildlife and farmers of Prydain for the last twenty years, increasing in numbers over the years. They suffer from an incurable disease similar to rabies, which has robbed them of all reason, fear and natural instincts. They live only for killing, rarely touching the meat of their killings. When they get the taste of blood, nothing can discourage them from the hunt. They attack and kill anything that moves — big cats, bears, entire wolf packs, any forest animal they can sunk their teeth into, livestock and humans (even a human village). Heck, they even attack the Horned King who strikes fear in every living creature.
Sonic X: Dark Chaos has Tsali. While moderately calm and even somewhat civil when he's not in battle, all bets are off in a fight. His greatest joy is murdering Seedrians in the most gory ways he can think of; everyone else is fair game too. And as the story goes on, the "Crazy" part comes to the forefront.
The Muslims are basically an entire faction of violently insane pedophile fundamentalists.
When Noahbrand appears for the first time he literally sears his handprint into Patrick's neck and seems to take unholy glee in watching Patrick burn. He does seem to be mostly Chaotic Good, though, seeing as he spared Patrick after learning about Michael.
Patrick, however, is a straight-up villainous example. He stalks Evan, seems to have creepy Foe Yay tendencies towards everyone in Team CTNE save Caught, and is quite frankly even creepier than he was in his original work.
In Thousand Shinji, Asuka is a rare heroic -or rather anti-heroic- example. After becoming a worshipper of War GodKhorne she was gradually turned into a psychopatic, blood-thirsty, axe-wielding berserker. However, although she went on a killing spree, she exclusively targeted thugs, rapists and drug-dealers, and she killed them because she was trying to save her boyfriend.
In the opening pages up Uplifted: Arrival Joachim Hoch ties noted real life Nazi War Criminal Oswald Pohl to his bed posts and proceeds to behead him with an axe. He then has the head sent to Berlin as a warning.
Forum of Thrones has Wolfius Woodbark, the psychopathic serial killer introduced early in Book 1. Over the course of the story, he shows absolutely no hesitation to act with utmost brutality, sometimes without any provocation at all. He is so unstable that his companions outright ditched him to make sur that he won't compromise their mission.
Kreep has signs of this as well. Most of the time, he shows no emotion at all, but his eagerness to get into a fight and the brutality he uss to dispatch his victims is unsettling, to say the least. Fittingly, he is revealed to be the son of Wolfius
Alysanne Waters is another example. She is as unstable as she is dangerous, a psychopath with a violent temper. One scene that sees her fighting against several men at once has her outright laughing at the violence around her.
Roar of the LION: Inori is violently unstable, and commits all manner of horrible atrocities and inhumane experiments, both For Science! and her own sadistic amusement. Even when working with Cinder and the White Fang, she regularly kills their minions for the sake of experimenting on their corpses, largely to use them as power sources for her machines. She's so depraved and monstrous that when Emerald and Mercury break into her lab on Cinder's orders, they're appalled by what they find inside; even Adam Taurus of all people is disturbed by her.