The Wayward creed in Hunter: The Reckoning are all Ax Crazy, even if they weren't before the imbuing. The trauma of the Wayward imbuing is such that the options are psychic annihilation or insanity; every Wayward who survived the imbuing came out insane. Furthermore, other Hunters show a preference to protect humanity from unruly supernatural forces; Waywards, on the other hand, are more likely to blow up any supernatural creature they encounter (no matter how harmless), and shrug their shoulders if any humans die as a result of "collateral damage." Making it perhaps even worse is that, despite their insanity, Waywards are planners, and quite, quite deliberate in pursuing their goals - specifically, the complete elimination of the supernatural. The biggest Ax Crazy is Joshua, AKA "God 45," who is implied to have stolen a nuclear weapon during the Time of Judgment. He becomes a player character in the video game Hunter: the Reckoning: Wayward, where Fanatic 656 displays the traditional Wayward traits.
Hunter: The Vigil, Reckoning's New World of Darkness counterpart, features slashers as antagonists, humans who are driven to kill. Some become mortal serial killers; others, however, begin to manifest supernatural powers to 'help' in their pursuit... The crowning example from that setting is the Hunt Club, a social club which turns serial murder into a contest.
Warhammer and Warhammer 40,000 both feature Khorne, Chaos God of mindless violence and senseless bloodshed. His mortal followers try to kill something every day, and have been known to take their own lives if no other targets present themselves - after all, Khorne cares not from where the blood flows, so long as it flows ("Blood for the blood god!"). Daemons of Khorne are the embodiments of rage and bloodlust, and number among the most terrifyingly effective warriors in the game.
Khorne's favorite daemonic servant is the Bloodthirster known as Skarbrand, who was manipulated by rival god Tzeentch into attacking Khorne himself. This ended with a furious Khorne grabbing him by the throat and hurling Skarbrand from his citadel so that he fell for seven days and lost his wings in the crash-landing. The experience left Skarbrand with no other emotion than pure, undistilled hatred and fury, and a mission to roam the mortal and immaterial universe and kill anything he finds in a vain attempt to appease his god - for Khorne is so impressed with Skarbrand's work that he prefers his rampage across reality. Skarbrand is so contagiously Ax Crazy, in fact, that any creatures nearby are also overcome by insatiable bloodlust, compelling them to tear apart their enemies in hand to hand combat.
Khorne's greatest mortal servants in the 41st millennium are the ChaosSpace Marines of the World Eaters legion, chainaxe-wieldingberzerkers who voluntarily undergo lobotimization to become more effective killers. The most famous and Axe Craziest of their number is Kharn the Betrayer, so named because he has a one in six chance of attacking any friendly models in close combat. Kharn is singlehandedly responsible for breaking the World Eaters as a coherent fighting force; when a battle against rival Chaos Marines ground to a halt due to an otherworldly blizzard, Kharn grabbed a flamethrower and torched every shelter, then went about taking the skulls of any soldier he found. As for Angron, the Primarch-turned-Daemon Prince of the World Eaters, Kharn was considered a calming influence on him.
On the subject of Chaos Space Marines, Dreadnoughts have a special Crazed! table they must roll on each turn, giving them a chance to lunge into close combat or unload their ranged weapons on the nearest squad, friend or foe. They're such demented killers that outside of combat, even after the pilot's sarcophagus has been removed, the empty war machine is kept chained to a wall, just to be safe.
Orks (and Orcs) are Choppa Krazy as a species, violent brutes who live only to wage war on anything around them or, failing that, each other.
Even among the Orcs, Grimgor Ironhide stands out. Other Orcs think Grimgor's a blood-crazed sociopath, and mostly only stick around him because he wins. If Grimgor goes without a fight for a day, he'll start picking fights with everything in sight. Two days, and he won't bother with the formalities, he'll just attack. His followers hurl themselves onto his axe rather than see what happens after three days.
Even heroic factions are subject to this. The Blood Angels' Death Company consists of Battle-Brothers who have fully succumbed to the chapter's flawed gene-seed, hallucinating themselves as their lost Primarch during the final battle of the Horus Heresy. Not only are they so consumed by bloodlust that they must charge into close combat, but their minds are so far gone that they Feel No Pain as well. They're not expected to survive their battle, and any that do are mercy killed by the chapter's Chaplains. The Minotaurs also have a reputation as vicious berserkers. Games Workshop has hinted that they were an experiment in using "purified" World Eaters gene-seed to create new Space Marines, which would make sense.
The Eversor Assassins of the Officio Assassinorium are infamous for being this. Calling them "assassins" is generally considered a bit of a stretch, as they are fully capable of decimating entire armies once released onto the field. This is primarily as they're hopped up on so many combat drugs that it's impossible for them to feel anything except bloodlust. And the Imperium designed them to be precisely like this. And should someone manage to kill them, they explode.
Eldar Exarchs are warrior-priests of Kaela Mensha Khaine, their bloody-handed god of war. Though the Eldar Path leads each member of the race to foster multiple personalities as they master one occupation at a time, the Exarchs are those who have become trapped on one of the Warrior Paths. Forsaking their past identities, they take up the mantle of the last to wear their armor and live only to wage battle on the Eldar's enemies. Biel-Tan craftworld is ruled by a council of such Exarchs, and known for its zealous and xenocidal campaigns to rebuild the ancient Eldar empire.
The Dark Eldar, in general. There is nothing they love more than the pain and suffering of others, they revel in carnage and death, and they have no control over their urges. Part of the reason they take part in realspace raids (besides souls and captives) is to relieve these urges on whoever gets in their way, and they don't care who these victims are; they view the entire galaxy as filled with prey organisms and playthings, basically. Given they're also Sufficiently Advanced Aliens who possess the technology to regrow and recover from most kinds of physical death, what would be a horrific life-or-death struggle for any sane human being is seen as nothing more than a bit of fun by them.
The Barghesi, a minor xenos race, are apparently so hyper-violent that they make Orks look well-adjusted by comparison. They're named after the Barghests, malevolent spirits of English folklore who often take the form of monstrous black dogs, and they live in the Grendel stars. Several Space Marine chapters are dedicated entirely to keeping them there, and apparently the Dark Eldar have taken interest in capturing them for Commorragh's fighting pits.
Even before they became the World Eaters, the War Hounds had a reputation for slaughter. The entire reason they were called War Hounds was because they reminded the Emperor of the vicious war hounds a warrior tribe on Terra used. Things got worse for them when their Primarch Angron took command of the legion. One of the first things he did was to have his Butcher's Nails copied and implanted in most of his Marines' heads. The Nails more or less enforced Ax Craziness since they made it nearly impossible for their recipients to feel joy from anything but rampant slaughter.
Cyborgs who get too many implants, in settings where Cybernetics Eat Your Soul. They're known as "cyberpsychos" in Cyberpunk 2020 and C-SWAT's special Psycho Squad gets called in when one goes on the loose.
In the fourth edition of the game, Gruumsh is changed from the axe crazy god of the orcs, to simply being the god of axe crazy-ness.
The Omnicidal ManiacGod of Evil Tharizdun is completely insane, and so are most of his worshippers. (Insanity is pretty much a prerequisite to worshipping him.)
In addition to Gruumsh, Forgotten Realms has Garagos, a practically nonsentient deity of bloodlust and destruction (materials that discussed Netheril and the gods indicated this was partly because that's just the kind of god Garagos was from the beginning, partly because he is really, really pissed off that the newcomer Tempus usurped the position of god of war from him, and partly because focusing on those aspects of war is what he did when he had to re-invent himself in a lesser niche after he lost the god of war position — the alternative to that last would have been bolting for another pantheon, except there wasn't one with a convenient hole around).
Surprisingly rare among darklords of Ravenloft, as most aren't truly insane, and hence are wholly responsible for their own evil actions. The Hive Queen, Tristessa, Malken, and Duke Gundar have all shown strong Ax Crazy tendencies, however, as do quite a few non-darklord villains like the Midnight Slasher. One Darklord who is very insane - which is no secret to his subjects - is Essan the Mad of Vechor. This is actually a benefit to him in many ways.
"Madman" actually had its own entry in the 2nd Edition Ravenloft Monster Compendium.
Even going beyond the Great Curse, it's possible for the Exalted to reach these lofty heights. Take Raksi, Queen of Fangs, who Exalted in her pre-teen years and was forced to flee into the Wyld during the Usurpation. She came out changed in several ways. Way number one? She took the city of Speremin by force and renamed it "The City of a Thousand Golden Delights," bred an army of ape-men herself, ran up a reputation as a cannibal sorceress queen amongst the local tribes, decided to cement it by making babies her favorite meal, and now spends her days desperately trying to learn the secrets of a style of sorcery way beyond her means and willing to do anything that will get her closer to revelation. The other way? Her fingers now bend backwards.
Amongst the Yozis, Adorjan may be the model of Ax Crazy. She views silence as a gift and loves everyone, so she wants to kill them all to show her love. Her Charm set for her Infernal subjects reflect this, including such options as "Social Influence attempts are blocked by all noise turning into a wretched discord that makes you want to kill" and "you can only verbalize laughter, but gain telepathic communication with allies."
A number of Limit Breaks, notably Berserk Anger and the Torment of Malfeas, send their bearers into a psychotic killing rage that doesn't end until everything around them is dead, on fire, or dead and on fire.
The Rifts RPG's Crazys are just as easily Ax Crazy as they are batshit insane.
Arkham Horror's maniacs. In particular, while most Ancient Ones have rules changing how the cultist enemies function, Azathoth's cultists are completely insane and count as maniacs.
In The Dark Eye the men and women who form a pact with Belhalhar, the demon of Bloodlust and Murder, tend to become this, if they weren't like this before, after a while because of the effects of the pact. The priests of the demi-god Kor act the same way, but they won't attack anyone who surrenders or is unarmed.
Almost all demons under the rule of Belhalhar act the same way, to the extent that getting them to stop fighting is quite hard.
The Maru, a race of crocodile people, almost all act the same way as directed by their god Kr'Thon'Chh; he and Kor are one and the same god, despite many humans' fear that the Maru follow a demon.
You get these from time to time in BattleTech. A few notables include Romano Liao, a violent paranoiac who took all her lessons in rulership from Stalin, apparently; Brett Andrews of Clan Steel Viper, known as "The Bloody IlKhan" for his almost wanton slaughter of political opponenets; Katherine Steiner-Davion, who murdered her own mother and brother and deposed two other siblings to ascend to power; Jinjiro Kurita, who ordered the massacre of an entire planet after a sniper shot his father on said planet; Thomas Marik proper, AKA the Master of Word of Blake, AKA the man responsible for the horrors of the Word of Blake Jihad... and that's the short list.
The defining trait of Heroes in Beast: The Primordial is that they are always, always a combination of this and Knight Templar. Heroes were never stable to begin with, but their exposure to a Beast drove them into obsessive psychosis, totally fixated on killing the Beast to the point they can — and always will — justify just about anything if it serves the pursuit of the Beast. Abandoning friends and family, whipping up frenzied mobs as sacrificial Cannon Fodder, burning down occupied houses; if it will give them some advantage against the Beast, then they'll do it.