Played straight, subverted, averted, and everything in between in Galilee. Although the Barbarossa get the most of it, the Gearys being more of the drunk-with-power type.
Luman is considered this at first, not helped by the narrator who doesn't trust him. The fact that he lives with lots of weapons, in his own piss and that his childhood is about getting to discover every nuthouse in the country, there's some reason to. It gets slowly subverted tough, as Luman shows more and more remorse and interest in his brothers, sisters and bastards. Late in the story he shows some intelectual acumen and helps Maddox accepting his past faults.
Cesaria gets the worst of it. Sure, she can show some genuine compassion and love... but no one who even remotely knows her forgets that she could kill anyone at any moment should she get angry. And she gets angry fast.
Late in the book, Mitchell slips firmly into it and it's hinted that Garrisson could end up the same way.
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.
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.
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.
The Belgariad: Taur Urgas, King of Cthol Murgos. 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, 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?
Phrygiar 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-defence" and suspected killings).
Rodya Raskolnikov from Crime and Punishment, is a study of how someone can become ax crazy only once (even if this once is to say Annie Wilkes levels). That was because he flipped out in the middle of a murder he'd already premeditated and his plan fell apart thanks to an unexpected witness which led him to panic and violently stab again and again, as it was now a matter of instinctive survival.
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.
Stygg from Dragons. He kills lots of families, just to replenish Grella's sewing supplies. He gets worse after ingesting dilute Ix.
Dying of the Light gives us the memorable Bretan Braith Lantry. Two-Faced, and rather touchy, he'd killed several people in duels before coming to Worlorn. Once there, he challenged an offworlder to a duel for touching his car, ranted in a "Little Dog Too" manner after getting into a second duel, killing everyone involved in the death of his teyn, and burning down an empty city. Then it turns out that he may be the least villainous antagonist.
In The Gods Are Bastards, the spirits that grant headhunters their power are utterly insane with rage and hatred, and need to be pacified with regular victims (ideally either challenging, deserving, or both).
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.
In Harahpin, Eyrco and Euron become this briefly when they first arrive on Untoria. If it wasn't for Euron's silver glow, Eyrco would have ripped out his throat.
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.
Some of the Careers in The Hunger Games. Clove would've given Katniss a Glasgow Smile if Thresh hadn't stepped in. And Cato explodes so violently when Katniss takes out his supplies that he snaps a nearby boy's neck. Enobaria ripped someone else's throat out in her Games. With her teeth. Titus tried to eat the hearts of the contestants he killed.
In Death: A number of the murderers Eve pursues certainly qualify as this. However, other murderers prove to be perfectly sane.
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.
Jeeves and Wooster: 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.
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.
The Bonehunters gives us sapper Crump, who is a firm believer in more is more when it comes to explosives, as Y'Ghatan's ex-city wall can attest. Unfortunately, the other Malazan soldiers rather adhere to the 'don't endanger your own troops unnecessarily' philosophy, wich means that Crump is not one of the popular guys.
In Toll the Hounds Amby and Jula Bole are just as creative in their use of weaponry and disregard for the safety of everyone involved as their brother Crump is.
In The Maze Runner, due to the Flare being a Hate Plague of sorts, those who are infected with it begin to show symptoms of paranoia, hallucination, and extreme, unwarranted violence: as they descend into madness and eventually pass the Gone, they become insane shells of their former selves, gruesomely attacking anything that moves and even resorting to cannibalism.
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!
To give you a real insight into just how dangerously unhinged Annie is, there's this one part in the book where she takes a rat she trapped in her basement upstairs and puts it in front of Paul. She then begins to crush the poor creature to death with her bare hands, all the while discussing her extremely nihilistic view of the world, with humans being merely trapped rats with broken backs. Then she takes the rat's blood and licks it off her fingertips.
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, which are all eventually revealed to be a result of the god of destruction whispering in his ear. The Inquisitors from the same series are also decidedly unstable, because they draw their power from the same god.
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.
The Mortal Instruments shows us Maureen. As a human, she is hardly different from other girls, but as a vampire she always massacred, and is glad about it. She is so obsessed with killing that she later becomes relatively easy to lure her into the trap.
Anton Chigurh from the novel No Country for Old Men. 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.
Nowhere Island University is a very dark story. As such, many characters, especially the ones from The Academy of Military Science and Shadowhaven, tend to be a little violent. However, there are some who deserve special mention.
Salim, much as he tries to justify it, seems to enjoy hurting people. Especially evident in Track 8.
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.
Renegades has The Detonator, a Mad Bomber with the ability to create explosives with her thoughts. She's also obsessed with harming Captain Chromium, to the point of endangering her teammates. Reality Ensues in that the moment this comes to light, the other Anarchists boot her out before she can harm them further.
Simon seems like a poised, cultured assassin at first, but as his Humiliation Conga kicks in and he repeatedly fails to complete a contract, the cracks begin to show. By the middle of book two he's willing to blow up over a hundred people, and possibly himself, just to kill a single target.
Mari's obsession with chivalry and knighthood is arguably the only thing that stops her from murdering anyone who looks at her sideways. When she's finally tormented to her breaking point, Nessa literally presses an ax into her hands and goads her into a killing spree.
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, smashed a baby's head against a wall before raping and murdering his mother, 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.
Joffrey Baratheon is an interesting example. He's a sadistic sociopath and has almost no regard for human life whatsoever, but he's also an abject coward who can't win a fight against a girl half his size, so rather than do the deed himself, he likes to have his thugs kill and beat people in front of him or shoot defenseless people with a crossbow from a safe distance.
Jaime Lannister considered the Smiling Knight to be his boyhood's Mountain, only "Half as big but twice as mad."
In Spectral Shadows Cygnusians can become infected with Red Vision, a genetic disorder that will make anyone this.
And then there's Lukas, who's psychotic and will kill anyone he doesn't like or who doesn't give him his way.
Joaquin from Star Trek: The Eugenics Wars is quite fond of throwing knives at people he doesn't like. Khan shows shades of this from time to time but he can control himself.
The Strange Case of Dr. Jekyll and Mr. Hyde — If Edward Hyde isn't Ax Crazy then 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 a random, 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.
Deca in Tales of an Mazing Girl seems to be a Large Ham of a villain in a vaguely-Iron Man suit. However, his actions tend to play a lot creepier and more psychopathic then a classic villain. He's 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.
Malus Darkblade from the Warhammer: Daemons 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.
A casebook sociopath, if ever he'd seen one. People called Larkin mad, but he wasn't mad like Cuu. Cuu was a cold killer. A psycho.
Mapleshade and Hollyleaf in Warrior Cats. Mapleshade is so messed in fact that the other villains are scared of her.
We Are Legion (We Are Bob): The Brazilian probes. They were all cloned from military men who are far more interested in destroying all competition than taking humanity to the stars. The Bobs adopt a shoot-on-sight policy with them, which proves justified more than once.
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.