Downplayed in Wolfenstein: The New Order. While much of it depends on your playstyle, it's made clear that B.J Blazkowicz is this-he's a bit too eager to go the extra mile in killing Nazis. Case in point, the fact that he can dual wield virtually every weapon in-game, he interrogated a nazi with a chainsaw, and stabbing people at the worst possible moment in cutscenes.
Xenogears has Id, also known as the Demonof Elru. Whenever he pops up, he goes around killing everything in sight, committing genocide, crashing a "space station" (killing all the people living inside), or just behaving like a "normal" serial killer, and murdering people one by one. In reality, Id is actually Fei's Split Personality. The guy has a lot of issues.
Captain Hayashi from Jet Set Radio will attempt to murder you for your petty crimes, sending in all the tanks and using a experimental Spider Tank
Catalina. In III, she already shoots two of her apparent love interests, but it goes even further in San Andreas: she has graves dug by her house, she sexually abuses CJ (and still thinks she loves him), she threatens to kill/castrate CJ, and she takes glee in killing any cops or civilians who get in her way.
T-Bone from Grand Theft Auto: San Andreas. He is brutally sadistic and impulsive when gets really angry. In "The Introduction" video, he gives a brutal beating of another band member, possibly to the point of killing him. He also threatened to rape and kill the family of CJ.
Ray Bulgarin from Grand Theft Auto IVand his sequel. While first referred to as extremely impulsive and aggressive tendencies, even worse when suddenly brutally shoots his sister, possibly killing her.
Some protagonists of the series also come here:
Tommy Vercetti from Grand Theft Auto: Vice City, is aggressive and easily angered, and has no problem want to kill their victims with chainsaws (not for nothing was nicknamed "The Butcher of Hardwood"). Also one occasion he was sent by the leader of the bikers to kill civilians and wreak havoc in the city.
Trevor Phillips from Grand Theft Auto V game is, to say the least completely fucking nuts. The trailer alone showed him beating people brutally and committing destruction for kicks, but that merely scratches the surface: He's completely unpredictable, extremely violent and easy to piss off, he's a meth addict and an alcoholic, he is heavily implied to occasionally eat people.
The Prophet of Truth from Halo, who in Halo 3 rants and raves and comes close to his goal of activating the seven rings that will kill all life in the Milky Way. While his portrayal in the novels and Halo 2 is more that of a cunning and sane manipulator, he was apparently driven to madness after the events of 2, when 343 Guilty Spark revealed to him the full truth about the Halo Array's purpose (though he already had his doubts to begin with).
Ripa 'Moramee, the Arbiter from Halo Wars, is an even clearer example.
Sapphire of Disgaea 3 qualifies. Not as dangerous as some here, but still quite violent.
Fritz from Brain Dead 13, whose deranged methods of killing involve eggbeaters, harpoons, and a blender. Combine that with him being a Determinator, and, well... you're screwed.
Arioch is an Ax Crazy Elven Action Girl. Due to the loss of her children (presumably right in front of her eyes and in some gruesome way), she's now a cannibal, and has a preference for small children. She's also on the player's side.
In Dwarf Fortress, every now and then one of your dwarfs may snap and start killing anything and everything in sight. The usual cause for this is their happiness falling too low, although there are others. Often from having seen/being attacked by another dwarf that had this happen to them.
Memorably, in the Something Awful Let's Play fortress of Boatmurdered, the fortress deteriorated to a level where it was in a constant state of anarchy, with dwarfs becoming so annoyed by the smoke and miasma that they snapped and went on a rampage, invariably killing a dwarf that was the relative of another dwarf, driving that dwarf into a homicidal rage, and so on. Probably the best individual moment of this was when one of the former governors of the fortress, who was possessing of insane strength, managed to beat several other dwarfs to death incredibly messily. DID I MENTION HE IS ON FUCKING FIRE!??
A (rather appropriate) meta example from that very Let's Play comes in the form of StarkRavingMad, one of the appointed overseers of the fortress, who devises a plan to finally end the siege of elephants that was threatening to kill them all:
StarkRavingMad: I've started project 'Fuck The World,' a top secret attempt to funnel magma to the outside. I'll kill those elephants. I'll kill all those fucking elephants.
For someone who ends all his conversations with "Visit again or I'll pluck out your eyes!", he is surprisingly nonviolent. Even if you attack him, he doesn't attack back directly, just scolds you and teleports you into the sky, causing you to fall to your death. Just don't make him too happy or talk to him while he's contemplating on having Brain Pie for lunch.
He's actually not technically in control of "Axe Crazy" only but Madness as a whole. Mehrunes Dagon and or maybe Boethiah might be more "Axe Crazy" since they're both a lot more violent and associated with violence. And well, actually having an Axe also used as a symbol of their sphere of influence. Dagon likes to sent you on a mission to kill people who he dislikes, if he's not trying to smash you himself.
Glarthir, from Oblivion. He believes three of the other NPCs are conspiring against him, and sets you to tail them. Not a single one of them actually is — but if you tell him this, he'll assume they've turned you as well, and whip out an axe to take you out before you get him. If you lie, he'll ask you to kill whoever you fingered, or do it himself if you refuse.
Krieg, the playable Psycho, deserves special mention for taking this trope literally.
Kefka from Final Fantasy VI. This is a man who reduced the planet to a burned-out shell because he could, and then randomly took out the remaining pockets of civilization with a Wave Motion Gun whenever he got bored. That's during his encounter with the Warring Triad. Beforehand, he was, while relatively normal, a man prone to breaking into fits of hysterical laughter when doing atrocious crimes, had brainwashed Terra and ordered her to torch fifty soldiers alive, attempted to burn Figaro, poisoned Doma (and in that particular case, he also mused maniacally about his joy about listening to "music" revolving around hundreds of people screaming in unison), brutalized various Espers in the Magitek research facility while maniacally declaring himself to be a god, massacring various Espers at Thamasa as well as burning Thamasa to the ground.
His backstory reveals that he was the first Magitek Knight, but the process was imperfect, and that he became insane after the process, which is heavily implied to involve brain surgery, meaning that the Empire broke his brain.
And in a vein rather similar to Kefka, there's BlazBlue's own Hazama/TerumiYuki. He trapped the world in 725 time loops of 100 years each through which the world, during just the first 10 years, got ravaged by an enormous, hydra-like monster. This is also arguably one of the less horrifying acts he has done, solely because the result of it was a continuously repeated impersonal and indiscriminate mass-slaughter on over 50% of the human race... Y'see, it's not until Hazama/Terumi gets personal with people that his maniacal depravity really shines through.
Carl Clover. Especially if you touch Nirvana/Ada, or try to hide information about his father Relius. In Tsubaki's story mode, losing to him leads to him killing his childhood friend.
Carl: Oh, please don't die yet, Miss Tsubaki...
Tsubaki: ... U-Ugh...
Carl: Well, if you'd be so kind as to tell me the whereabouts of my father... Preferably before you bleed to death.
Jade Empire's Black Whirlwind is a hot-headed hulking brute who carries twin axes, has killed innocent noblewomen, burned down temples, and more, most if not all while drunk, but he's on your side.
Ignus, a fire mage, from Planescape: Torment, who nearly burned down the city of Sigil before being stopped, can join your party. Ravel Puzzlewell, who almost destroyed the city of Sigil before the Lady intervened, is connected with your character. The Xaosects/Chaosmen, represented mainly by the insane Starved Dogs Barking thugs, prowl The Hive (though you can speak to one, Barking-Wilder, and turn him briefly sane). Acaste, the Ghoul Queen, hungers for flesh, though that's pretty standard procedure for a ghoul. One of your own past incarnations was quite ax crazy and damn near impossible to kill, to boot. You can use a fossilized arm of his as a weapon.
The crown king of all Ax Crazy RPG villains is Luca Blight from Suikoden II, hands down. Nothing is safe from his wrath...not even small children. In fact, Luca rewrites the book on Bad Ass, as it eventually takes as many as eighteen of your party members, several volleys of arrows (each one shown capable of immediately killing one of his soldiers), and after all this, a duel to the death with the main character to bring him down in a suitably epicClimax Boss battle.
Albedo from the Xenosaga series. The particularly disturbing "Ma-Peche" scene gives testament to this, especially when he tears his own head off to prove a point about how he can not die.
In the Japanese version, he cuts it off, but the edited scene in the English release actually manages to be more disturbing than the original.
Omega from the Mega Man Zero series is a mechanical embodiment of this trope. His sole purpose is simply to kill. His bloodthirsty nature rivals even a Khorne Berserker.
Worse than that, he's exactly what Dr. Wily had intended the real Zero to be. In fact, Zero is indeed Ax Crazy, before his Heel-Face Turn.
Omega's own creator, Dr. Weil, also qualifies. Destroying a whole neighborhood just to capture the Dark Elf? Check. Making humans' lives completely miserable, with the choice of either dying or living under his rule? Check. Setting a Kill Sat to destroy a human encampment? Check. Plus, going completely nuts after the Kill Sat is set and fired against himself, initiating a Colony Drop with him inside the satellite, believing himself to survive the crash...
He's such an incredibly dangerous person to have around that even after his bodily death, his sheer loathing and insanity leached into the surrounding wreckage, giving inanimate material the ability to feel hatred and desire complete obliteration of all sentient life on the planet. Even so killed and diffused, he's still so thoroughly evil that mere proximity to fragments of this wreckage can drive a person to insanity, which eventually becomes a major plot point in the sequel series Mega Man ZX.
Vile from Mega Man X. Similar to Galvatron, an accident regarding his brain led to him developing a disturbing relish in not only terminating Mavericks, but also causing as much collateral damage as he can while destroying the maverick in question. He also has a sociopathic hatred of X, and he revolts against authority namely to cause destruction.
The sequel invokes this again by making it look as if Clover killed Luna in a rage. Subverted, though, as it was an accident.
Super Paper Mario has Dimentio. It's not obvious at first, but about three-quarters of the way through the game he starts killing characters left and right, all the while cracking jokes and delivering lines like "It won't be so bad, I promise", in addition to being The Starscream. He's also referred to as being psychotic at least once in the game.
Colonel Volgin of Metal Gear Solid 3: Snake Eater. Highlights of Volgin's psychotic dickery include nuking his own country, trying to destabilise the Cold War, and randomly torturing and killing his own minions simply because he can. When Snake's eye gets shot out, he practically orgasms at the sight of it.
Sundowner, Mistral and Raiden himself in Metal Gear Rising: Revengeance. Sundowner is a war-loving murderous sociopath who uses scissor swords, Mistral butchered the murderers of her parents and arranged for Khamsin's death simply because she found him annoying, and as for Raiden, it's revealed that he always was somewhat Ax-Crazy ever since he was a child soldier, but only after a Breaking Speech by Monsoon does his darker side resurface.
Daniella the maid from Haunting Ground. She is a homunculus created to be the "perfect woman" by her master. However, she cannot feel pain or pleasure, which drives her to scream at her reflection because she feels she is not "complete". She becomes even crazier and hateful towards Fiona, who can feel what Daniella cannot. Daniella comes to a logical conclusion - she must obtain the vital essence known as Azoth, which Fiona possesses, in order to become complete. She plans on tearing the Azoth right out of Fiona, and pursues her relentlessly throughout part of the game. She is prone to fits of maniacal laughter (her head often spasms violently when she does so) and will shriek at random.
In the first MechWarrior, you can get pilots to help you in your group, each comes with a picture and a quote along with their stats, before you hire them. One of them is Killer, wearing a ski mask, with the quote "I want to burn villages and eat dead, burnt bodies, kill, kill, kill!"
The quote is from the 1969 movie Alice's Restaurant. Arlo Guthries character acts like he's crazy in order to get out of military service. But despite his best efforts he seems to be sane enough for them.
He gets an homage in MechWarrior 2: Mercenaries in the form of Jake "Killer" Ives, a cheerful psychopath who was also a cheap, effective pilot, noted in his hiring profile to be a sadist bar none, with Humongous Mecha as his tool of choice for hurting other people.
Many of the bosses in No More Heroes, and more so in its sequel. Special mention goes to Bad Girl, who bats blindfolded gimps at you and beats you with a burning baseball bat, the insane superhero Destroyman, the pig-tail wearingYandere Kimmy Howell, the damned soul of Matt Helms, and Jasper Batt Jr. after taking some Psycho Serum. In fact, most of the remaining assassins come off as Retired Monsters who got Bored With Insanity.
From The Mark of Kri we have Tati. This deserves special mention because her older brother Rau is the one with the ax, but he's remarkably calm and level headed. And he never speaks in anything except battle cries. Unless Tati turns to The Dark Side, in which he will utter her name once before Tati and the forces of darkness slaughter him. Tati, on the other hand, is very vocal and has moments of exceeding maliciousness. The way she slaughters single enemies with her knives looks more painful than anything Rao does with his ax, and considering the body count Rao racks up with it, that's saying something.
Barbatos Goetia of Tales of Destiny 2 is also an example of Ax Crazy trying to prove that he's the strongest out there, which caused him to be written out of history in the first place. It's not just his personality that is insane, his power is also insane that he can even be invulnerable against magic whenever someone tries to cast one. He also wields an ax. If you want more of his Ax Crazy stuffs, just listen him pull an Evil Laugh or his Kiai.
The main reason Jade Curtiss from Tales of the Abyss banned replicating living creatures was that when he attempted to revive his mentor who he accidentally killed through it, the resulting clone was an ultra-powerful Bonus Boss with a sociopathic personality. No wonder he was a bit apprehensive about using it again.
Zagi from Tales of Vesperia, who, at about 3/4 through the game seems to have no reason for wanting to kill Yuri other than that he wants to kill Yuri. His Ax Crazy status is lampshaded by his one eye that's smaller than the other possibly because he's so insane that he can't be bothered to hold them both open. In one of the boss battles against him, he casts poison on HIMSELF! Most battles consist of him spouting random crap about wanting to kill Yuri.
Agria from Tales of Xillia. Her looks, posture, and frequent laughter make it readily apparent that she's got more then a few screws loose.
Jerme, the Black Fang assassin nicknamed Death Kite from the same game, fits this one like a glove. He says to Lyn: "You! Woman! You have beautiful skin. If I sliced you into lovely red ribbons with this, would the pieces be as soft and delicate as silk?"
The Scout considers hurting people to be his purpose in life.
The Soldier, after being refused entry by the army during early WWII, took a ship to Europe and fought "the nazis" until 1947.
The Heavy really likes fighting, and sometimes takes advice from inanimate objects ("What's that, sandvich? Kill them all? Good idea!").
The Pyro is a literally Ax-Crazy Psychopathic Manchild who sees the world as a Sugar Bowl full of giant lollipops and infantile cherubs. When s/he's killing people with a flamethrower, from his/her point of view s/he's just blowing bubbles and rainbows at them.
And there's the ax-wielding Horseless Headless Horsemann, the NPC from the Scream Fortress update, who attempts to hunt down and behead every player he encounters.
Needles Kane the driver of Sweet Tooth in the Twisted Metal series, especially his Twisted Metal Black incarnation.
The cast of Twisted Metal Black is made up of delusional priests, hillbilly psychos, and doll faced murderers. The sanest characters in the whole thing are Calypso, Agent Stone, and John Doe. The last two are genuinely good and sane people, who are in it for just reasons, and were institutionalized for severe PTSD and amnesia-related conditions. Still, the fact that the Big Bad is saner than most of the cast is somewhat... disturbing.
Aran Ryan from the Wii version of Punch-Out!! Most of the other characters emit grunts of pain or anger when you've landed a massive combo on them. Him? Deranged laughter.
In between rounds, he shouts to Mac, "Keep hitting me! I love it!!" Then, when he gets back in the ring, he punches himself several times, then makes an insane smile at the camera. His fighting style involves him running around the ring and attacking from the side, as well as jumping off the wires and trying to punch. This guy was perfectly normal in his first appearance.
"Your wimpy little punches won't even faze me." There's taking a hit and then there's getting your jollies from it.
Bald Bull from the same game gives a similar impression, both in that before matches he'll beat himself over the head for no apparent reason, and the way in which he rises from a knockdown in Title Defense mode is downright terrifying.
In the sequel, HK-47 is back again, and it's possible to pick up a truly psychotic Wookie named Hanharr. Atton is also implied to have more than a little of this. Much more than a little, if you encourage it...
Batman: Arkham Asylum has Quincy Sharp, who is by all appearances normal when you speak to him except in the Spirit of Arkham, where it can be seen that he is an absolutely insaneKnight Templar.
The main villains of the plot, some of the absolute craziest of Batman's rogues gallery. Zsasz, Joker, Killer Croc...
And of course, anyone who has been injected with some form of the Lazarus Vector: Lazarus Elixir, Lazarus Phoenix Pinion, Venom, or Titan. Who made these potions and how they are interconnected in the series is a major spoiler, but is responsible for about half of the plot. And yes, Batman also took it.
In a strange way, the player character makes Batman crazy. What else would you call gliding around Gotham City and Arkham Asylum looking for gang members to horrendously punch, explode, electrocute, freeze, glue, hang, and humiliate, all while leaving them alive and in tremendous pain / fear? For "EXP" so that you can equip Waynetech that Batman already has? OR leaving criminals to do their thing while you scour the map looking for trophies that look like green hooks?
The Burning Blade clan in Warcraft II and World of Warcraft deserve a mention, as a clan of Ax Crazies that were held in check by a large group of ogre enforcers. The Horde "was not willing to employ them except in the most extreme cases, for fear that they might turn on other clans or even each other".
Though decidedly one of the less maniacal cases of insanity, it is heavily implied that the Ghosts in the StarCraft universe would've liquidated the universe if it weren't for the inhabiting implants in their brains.
Whenever I see a world untouched by war, a world of innocence, a world of lush forests and clear rivers... I really just wanna nuke the crap out of it!
Zombies Ate My Neighbors has both Chainsaw Maniacs and traditional ax-wielding Evil Dolls. The sounds of a chainsaw revving offscreen coupled with insane children's laughter serve to keep you awake at night.
The Big BadElizabeth Greene from Prototype is so Ax Crazy that she forces Alex Mercer (who is quite violent in his own right) into an alliance with Blackwatch in order to bring her down. The fact that her mind was fried by Carnival II some time ago might have something to do with this.
Both the game and manga of World Destruction has the heroine, Morte, who is more than happy to blow up a city. The anime version, however, gives us the resident chick Rhi'a, who has no qualms to pull out her guns when she is pissed.
The Darkspawn of Dragon Age are a race of Ax Crazy Blood Knights. Being the embodiment of The Maker's condemnation of mortals (according to The Chantry), the Darkspawn exist to kill and do horrible things to the world. They will do their best to kill anything and everything they come across that isn't also Tainted; if they can't find anything nontainted nearby to kill they will readily try to kill each other. Only an Archdemon has the power and will to unite the murderous bastards and keep them from killing each other when they get bored. Unfortunately, Archdemons, being Tainted dragon gods, are also Ax Crazy. That's why Blights happen. Even some of the "freed" Darkspawn in Awakening are Ax Crazy.
Mortal Kombat has a few characters known for being lunatics. Not counting brainwashed victims and near-mindless beasts, the ones that stand out most are Baraka, Mileena, and Shao Kahn.
Probably Skarlet too. Never mind the fact that what she does to her opponent is violent even by the standards of this game (sort of justified, as her powers are supposedly derived from absorbing the blood of others) she seems so incredibly unstable that Shao Kahn had to keep her in chains when he didn't need her to fight or kill someone. (And despite that, she still showed Undying Loyalty to him; of course, she was one of his evil creations from the Flesh Pits, like Mileena, who also was incredibly loyal.)
GLaDOS from Portal is less ax crazy than she is coldly and calculatedly crazy, but it ends up having the same net effect on the people she murders. Supplemental material for Portal 2 indicates that, at first, the researchers were only able to turn her on for a few fractions of a picosecond before she tried to kill everyone. So, calculated murder is actually an improvement.
And Yuuka Kazami was this in the PC-98 games. Genocide was just a game to her... It's been dialed back quite a bit for the 9th game in the series... Though she is still able to worry some characters, including Cirno of all of them. And there's very little that worries that ice fairy even slightly.
And now from Word of God, Kisume has been confirmed as this much to the fans surprise. This also makes her somewhat unusual for this trope as she is also a Shrinking Violet.
Majora from Majora's Mask. Unlike most of the other villains in the franchise, it's main motivation for killing/tormenting/etc. its victims is for sheer fun and tries to take it to its logical extreme by crashing the Moon into Termina.
Zant from Twilight Princess, as shown with him killing the queen of the Zoras, enslaving the Twili against their will, and his unnerving confrontation with Link and Midna at Lake Hylia, where he tries to kill the latter. By the end of the game, the craziness fully manifests itself as he's reduced to a screaming, maniacal wreck.
Nikolai Diavolo in Everything or Nothing. Though he never shows this trait within the game, M reveals that even the KGB tried to kill him. To make matters worse, his mentor was the VERY ax-crazy Max Zorin.
Saints Row: Johnny Gat in 2, the Boss him/herself in 3.
Lilarcor the talking sword in Baldur's GateII. It's a fairly powerful sword that houses the soul of a bloodthirsty idiot. Every bit of Lilarcor's dialogue consists of it whining about how its not killing anyone at the moment and asking you to hurry up and kill someone with it.
Dark Souls has Lautrec, who is a loner vagabond who uses and kills Anastacia, the fire keeper of Firelink Shrine. The Darkwraiths you meet in game became Ax Crazy after becoming Drunk with Power and undead that have lost all their remaining humanity are insane, violent shells of who they once were. That said, some hollows you find are docile and often huddle themselves into corners, crying.
A great many people in Far Cry 3, which the advertising for the game heavily emphasized.
Special mention goes to Vaas Montenegro, who's crazy even by everyone else's standards.
Street Fighter IV's Juri is probably the nuttiest character in the bowl that is Street Fighter. Psychotic with a heavy dose of Sex Is Violence, Juri is quite the unchained animal when left to her own devices.
BioForge: The victims of the cyborgization process kept becoming this until the Mad Scientist in charge found out how to stop this. (The player also has the option of acting Ax-Crazy themselves, as there are points where you can kill someone even though they pose no threat to you.)
OFF has both Japhet, who's been driven from perhaps the nicest guardian to a lunatic whose policy became "worship me or die" after a couple years of thanklessness, and any Elsen that's been deprived of their sugar for enough time, who will assault even the spectres if they dare be in the same room.
The Splicers in BioShock have been reduced to this from repeated use of the Psycho Serum ADAM. Later you learn that they've been targeting you specifically because the ADAM they've been using has an additive that makes them susceptible to Mind Control by Andrew Ryan. If it weren't for that, they'd probably going ape on what's left of each other as well. By comparison, the Splicers in the sequel seem to be in better control of their faculties, if still mentally unstable and in the habit of constantly talking to themselves; the Spider Splicers may be an exception.