FBI Agent Pynebox in The Adventures of Fox Tayle does this after he breaks off from the rest of his team. He uses a gun, a knife, his fists, and a switchblade.
Sallie Declan, the Villain Protagonist of A. N. Wilson's A Jealous Ghost, has longstanding...issues. While babysitting a six-year-old boy, she loses her temper and hits him on the side of the head, hard, sending him smashing into a faucet. Later on, in college, she loses her temper again and nails a fellow student with an iron. And, making the third time the charm, when Sallie is dismissed from her job as a nanny, she murders a young girl by smashing her skull in and slashing her face to bits.
Visser Three, the Big Bad of Animorphs. Even in an alien empire where killing your subordinates seems to be routine, he's The Caligula. At one point Visser One remarks that he's executed subordinates 'by the poolful', which means that his body counts runs in the high thousands. At least.
Taylor. And probably Rachel. No, Ax is not Ax crazy. (He may act crazy around certain foods, though.)
Gregor Clegane, who, when he was a teenager, burned off half the face of his kid brother for playing with one of Gregor's discarded toys, is implied to have murdered his father, younger sister, wives and multiple servants, and who spent the majority of the War of Five Kings leading a band of Sociopathic Soldiers in raping, burning and murdering all across the Riverlands. You know you're insane when even The Hound is scared of you.
Interestingly enough, Arya Stark (a survivor of one of Gregor's murderous rampages through south-central Westeros) seems headed in this direction. She's one of the good guys, though. It's a real pity that she never got the chance to give Gregor a taste of valar morghulis ( High Valyrian for "Everyone Dies") before his death.
Belgariad: Taur Urgas, King of Cthol Murgos. Whoo boy. This is a man who beats his wives, Bad Bosses his underlings, tortures his enemies slowly, encourages his children to murder one another in order to become his heir, froths at the mouth in battle, chews the furniture during fits of madness, and dies screaming for his opponent to "come back and fight!"
Odiana, a recurring antagonist in Codex Alera, is a powerful watercrafter driven mad by being raped just as The Empath part of her watercrafting was coming in. She is quite nonplussed about physical violence, at the very least.
Odiana: If you go and kill the ugly little girl right now, won't the steadholder object? And then you'd have to kill him as well. And anyone else upstairs. And all these people here... Why shouldn't we do this again?
Phyrgiar Navaris, an antagonist in the fifth book Captain's Fury, is obsessed with becoming known as the greatest blade in Alera and has an official kill record in the three-digit range (potentially four if you include "self-defense" and suspected killings).
Thief of Time has multiple ax-crazy moments, such as Jeremy Clockson's behavior when he hasn't had his medicine, and Mr. White's mental breakdown while holding an actual ax.
Discworld's greatest example of a hero who has the potential to go Ax Crazy is Samuel Vimes, Commander of the Ankh-Morpork City Watch. All who know him are very afraid to make him too angry for fear that he'll snap and (to use the British term) "go spare". His most notable instance of rage, as seen in Thud!, made a group of dwarves, normally trained to fight to the death, flee in terror. To his credit, Vimes remains able to escape his rages before going beyond the point of forgiveness.
The Composer of the web-novel Domina considers it an act of admirable willpower when he doesn't murder a prisoner just because he can. His larger motives remain unclear, but its looking like he doesn't care for anything besides For the Evulz.
Terra from Patricia McKillip's Fool's Run killed over a THOUSAND people with a laser rifle in a single rampage....though she's surprisingly calm whenever encountered later.
A number of villains in books by Tom Holt display this. For instance, there's the evil genie in Djinn Rummy, or Jupiter in Ye Gods, who plans to wipe out the Earth and replace it with a nearly identical one. Neither one is very rational.
Drake Merwin from the Gone series, especially once Sam burns his arm off and gets it replaced by a snake-esque whip arm. And even worse now that he's back from the dead and Sharing a Body with Brittany, who Sam won't kill because she's innocent.
Also Caine after his visit with the Darkness in Hunger.
Gaia might as well be the poster child for this trope.
Bellatrix Lestrange from Harry Potter, whose love of Cold-Blooded Torture exceeds even Voldemort's. In fact, there are numerous occasions where Voldemort decides he has to stop Bellatrix from killing anyone (admittedly because killing them would ruin some plan of his, rather than out of any actual mercy, but even so...)
According to the narrator of Hell's Children, by Andrew Boland, Acheri is one of these. In that moment, when survival is the only thing worth thinking about, only a fool would tax there mind with grievances of the past and Acheri was no fool, axe crazy, yes, but no fool.
Brinkley, the replacement Bertie hires when Jeeves quits in Thank You, Jeeves, is a Battle ButlerGone Horribly Wrong. Ordinarily he's merely unpleasant and creepy, but when he gets drunk, he chases everyone relentlessly with carving knives, choppers and potatoes. And, since Bertie never actually gets near enough to fire him, there's some Fridge Horror inherent in the fact that he's still on the loose, and may at any moment decide to take a stab at being Bertie's valet again.
Stygg from The Last Dragon Chronicles. He kills lots of families, just to replenish Grella's sewing supplies. He gets worse after ingesting dilute Ix.
Roger and Jack of Lord of the Flies eventually become this during their stay on the island. Early on Roger was tormenting the "littluns" for fun, throwing rocks at them. In the end, Roger pushed Piggy to his death by rolling a massive boulder on him. Jack used him as a torturer, and he even made a stick sharp at both ends in order to impale Ralph and roast him over a fire.
No female fits this description better than Annie Wilkes of the Stephen King novel Misery. She started out as a mercy killer, killing babies who were not expected to live due to poor birthing. When she was fired from her nursing job, and she was no longer able to perform her 'public service,' she became absolutely bonkers. She has a scrap book containing news articles relevant to her murders, kidnaps her favorite author to force him to write a new book, resurrecting her favorite character which the author had killed (She treats this as if he had actually killed a real person), breaks the author's ankles with a sledgehammer to prevent his escape, and murders a sheriff who discovers her deeds. She's so horrifying, that even after the protagonist has bashed her head into a fine paste, he can't convince himself she's dead. He sees her every where he goes. She was so crazy she was contagious!
While in the movie she breaks his ankles, in the novel she goes a bit further by chopping off one of his feet with an axe, and then cauterizing the wound with a blowtorch.
Zane of Mistborn is a self-admitted lunatic who hears a voice in his head every time he sees someone except for the heroine telling him to kill them. Normally he does his best to ignore it, but every so often he will kill or maim someone (or just cut himself) to keep the voice under control. He's also psychotic in other, more subtle ways and in fact his violent insanity was what allowed the voice, actually that of the literal god of destruction to speak to him in the first place. The Inquisitors from the same series are also decidedly unstable, because they draw their power from the same god.
Not quite. Zane had a hemalurgic spike which enabled him to hear the god. No one can hear the god without one, no matter how insane they are. When Zane dies Ruin finally informs him that he was never really insane at all.
In Mogworld Mr. Wonderful was already like this, and the way for the past few years everyone who dies respawns at a church seems to have made him a lot worse. At one point in conversation he cleanly severs his own hand, keeps talking until he dies of blood loss, respawns at a nearby church, and walks back in to continue the conversation while casually gnawing on his own corpse.
Anton Chigurh from the novel No Country for Old Men is Ax Crazy personified. He has almost no personality other than pure murderous evil, which is made abundantly clear throughout the book.
He's actually more of a subversion as it is stated that he does have a personal code that he lives by. Unfortunately, the code he follows only makes sense to him alone and doesn't make him any less evil.
Erik from The Phantom of the Opera certainly has more than a few Ax-Crazy moments. His unfortunate habit of garroting anyone that sees him, aside from Christine, is certainly proof of this.
Please Don't Tell My Parents I'm a Supervillain: Jagged Bones, one of the villains hired by the Council of Seven and A Half to teach the Inscrutable Machine a lesson. His response to Claire's super-cuteness was to decide he wants to own her skeleton.
Richard Lopez of Ship Breaker, a drug-addicted nut who tries to kill his own son.
The Strange Case of Dr. Jekyll and Mr. Hyde — If Edward Hyde isn't Ax Crazy I don't know who is. He does pretty horrible things throughout the book, but the most gruesome of his deeds is the murder of one of the leaders of parliament. He has completely no reason for it, no provocation, and bashing in the face of old gentleman isn't really anything that sane men would do. As described in the book:
The spirits of Hell woke up in me to their full rage. I hit the defenceless body within intoxicating ecstasy and I enjoyed every single hit. Then I ran along the lit street still full of this amphibious euphoria, gloating about my crime and at the same time planning lightheadedly new ones, although listening carefully in case for the steps of chasing avenger. With a song on his lips Hyde mixed the potion, and upon drinking it he drank the toast of dead man.
In Spectral Shadows Cygnusians can become infected with Red Vision, a genetic disorder that will make anyone this.
Deca in the Tales Of An Mazinggirl seems to be a large ham of a villian in an vaugely iron man suit. However his actions tend to play a lot creepier and more Pyscopathic then classic villian. Hes not above killing people-not to make a point, but to make a point that there is no point.
Francis Begbie from the book (and The Film of the Book) Trainspotting, despite being one of the few members of his group to not use heroin, is the token berserker psychopath of the story, once casually injuring a random woman by throwing his beer mug off of a balcony and hitting her in the head, just so he could start a massive Bar Brawl.
People called Larkin mad, but he wasn't mad like Cuu. Cuu was a cold killer. A psycho.
Malus Darkblade from the Warhammer: Deamons Curse series is a shining example of Ax Crazy. He eats the heart of a previous captor, makes his oldest brothers face into a mask, and murders his father for a knife. And that's before the deamon stole his soul.
This is not in any way unusual in Druchii culture, for example, that father of his tortured him for about a week for some reason. And he really, really needed the magic knife, and didn't know who had it until he barged in and started killing people about it. The others had also tried/succeeding in doing rather nasty things to him in the past.
Mapleshade and Hollyleaf in Warrior Cats. Mapleshade is so messed in fact that the other villains are scared of her.
In The Wheel of Time, all male Channelers eventually either go Ax Crazy or die of a hideously disfiguring disease, thanks to a curse levelled on them by the Dark One (and the ones who do go crazy still die eventually- it's a toss-up what order the symptoms manifest in, how strong they are, and when). From the same series, recurring villain Padan Fain is also like this- being an imperfect merge of the original Fain and the ancient, malevolent spirit Mordeth he's about as far from stable as you get and is prone to lashing out violently at anyone unfortunate enough to be in his vicinity, though his real target is Rand. Scarily, he can still be charming (in an oily sort of way) when he wants to be, and has even shown the ability to supernaturally infect those he spends time around with his own Ax Crazy.
In Rand in particular this progresses to the point that Rand refuses to admit he feels anything at all, leading him to balefiring Natrim's Barrow out of existence without even blinking about it. Rand thinks balefire kills someone forever, removing them from the eternal cycle of death and rebirth. His only reaction was to double check that his target had been killed. He nearly killed his own father a short time later for trying to help him.
Subverted in World War Z. T. Sean Collins is a mercenary who lives to kill, and seems incapable of stopping himself. He was a Private Military Contractor before World War Z, and switched to zombies when the war broke out. The subversion comes in the fact that he is deeply introspective and well-aware of his mental state, and under no illusion of being able to give up killing. He states an intention to go on hunting zombies for as long has he can, and then killing himself so he will never risk relapsing into killing humans.