Mortasheen as a setting has plenty of this, but one of the most notable examples is Harlequeen, which is noted to be blitheringly insane even by Joker standards, but their plans turn out as if they had been calculated by a tactical genius.
Every popular Ork, and half the popular characters period, in Warhammer 40,000. Crazy and awesome are both plentiful in this setting, but Orks really excel at combining the two. They're so crazy that their technology is fueled by Insane Troll Logic and STILL manages to work... somehow.
He once took out a Titan by ramping his bike into it, bypassing its force fields (which set him on fire, by the way). He carries the crew's skulls with him and to this day, they're still on fire.
Ork flyboys are orks whose obsession with speed is considered insane even by other orks. To quote the deffkopta's inventor, Kog da Flymek, "Wot's faster than a warbuggy, more killy than a warbike, and flies through da air like a bird? I got no bleedin' idea, but I'm gonna find out."
Boss Tuska the Daemon Killa; a Warboss who decided that the Warp sounded like a fun place to be, so he mustered an army and charged into the Eye of Terror, slaughtering every daemon he could find. The orks were finally defeated on a Living Planet, but not before Tuska dispatched the offending Daemon Prince (who had just slaughtered the last of Tuska's weirdboyz with a gesture. Tuska responded by reaching between the daemon's legs with his power klaw and making a gesture of his own). The best part is that every day following that, the orks are resurrected and forced to fight impossible odds... meaning that Tuska found the ork equivalent of Valhalla.
He single-handedly reorganized his Legion in a single night. When the World Eaters stopped their assault on the Emperor's Children due to the extreme cold, he ran around with a flamethrower torching down shelters and Chaos Space Marines alike. When you serve the Blood God, you do not pussy out because of a little subzero temperature.
In older parts of fluff, Kharn actually had to be killed in order to get his ass off the battlefield during the Siege of the Emperor's Palace. He later woke up again, rage and all, furious at having been carried off the battlefield (by that point they had already left the planet, making it too late for Kharn to return). It's implied that either Khorne revived him, or Kharn awoke through sheer force of will.
Ork freebooter Kaptin Bludflagg. An Inquisitor tries to recruit him to serve as a distraction to the other factions in the sector. He agrees... but only if he gets to keep her hat. He gets it in the end.
His first mate Mista Nailbrain asks if they can keep a battlewagon (named Daisy) they just destroyed. "Okay Nailbrain, but it's yer charge! Keep it fueled and armed, and take it out fer rukks."
Hell, his whole crew qualifies. The aforementioned Mr. Nailbrain keeps the aforementioned battlewagon, and has a radar-like thing device he calls his "gitfinder", which can be apparently be set to "Panzee", Spookums is one of the few sneaky orks you'll ever (not) see, despite carrying enough explosives to level several buildings at once and not being afraid to throw about half of them at anyone he ambushes, and once tried to hide in lava, and Brikkfist has a penchant for strapping huge rockets to his back for a chainaxe-assisted Dynamic Entry at will, and would gladly use one of the Imperium's planetbuster missiles for this if he could get his claws on one. How effective is this bunch of loonies? They take down a centuries-old ascended marine, who used to be a Chaptermaster, an Eldar craftworld, a good chunk of several legions, among others, leaving a good part of the sector in ruins, and loot a gigantic Space Hulk, taking it as their new krooza and teleporting away to get themselves some new victims while Aurelia's still burning. And might we remind you, the Kaptin did all this for a hat.
Basically in the Warhammer or Warhammer 40000 universes, only two kinds of people can consistently survive more than two seconds: Knight Templar, and Crazy Awesome.
182. No figuring out the plot and killing the actual villain five minutes into the adventure. 337. Even if the rules allow it, I cannot control 20,000 pigeons and use them as flying piranha. 598. Any adventure that ends up with my character being worshipped as an orc god was just a dream. Retroactively if need be. 680. My axe doesn't go off accidentally when I'm cleaning it. 806. My character cannot have a noticeable impact, positive or negative, on a town's population. 1273. Any character that makes a seasoned Rifts player flinch is vetoed, and shall never be spoken of again. 1317. My character will refrain from appearing with Hitler in any history books. Especially if I'm chasing him with a wheat thresher. 1411. Despite what the rules say, bobsledding through the Vatican is much harder than it looks. 1939. Even if the rules allow it, can't parry an artillery barrage with my fists. 1951. Can't use the international date line to get around once per day restrictions.
To give two examples, the first playtest game ended up with Lemurians in a flying giant metal squid attacking a Nazi island by draining a giant emerald shaped like Hitler's head. The second playtest game ended with the PCs fighting Nazis on top of a blasphemous pyramid in Jakarta, with one character directing temporally displaced Persians from atop a jury-rigged flying machine and the other going toe-to-toe against the Big Bad to stop him from sacrificing the PC's sister to release a Timeline eating True Fey.
Pretty much every incarnation of The Loonie player archetype attempts to be this, although the balance between crazy and awesome varies.
Exalted. It has two martial arts dealing with flamethrowers and another that uses a chainsaw version of Captain America's shield. The main characters are assumed to go Serial Escalation on a daily basis. The default starting character types, the Solars, have to deal with fate ninjas who don't like them, elemental-powered samurai monks who really don't like them, and extradimensional living cities that absolutely hate them. Oh, and the mechanical spiders that control fate have a soft spot for feats of epic awesomeness, making the really insane stuff more likely to succeed. Given that it has been described as Dungeons & DragonsmeetsTengen Toppa Gurren Lagann, this was frankly inevitable.
The Ink Monkeys blog has produced some new enhancements. References to a First Age Solar setting up a special location for using tyrannosaurs as skis are just the start.
In the Pathfinder Campaign Setting, there is the deadly Test of the Starstone, which turns any mortal who passes it into a god. The adventurer Cayden Cailean went on a roaring drunken bender one night and took it on a dare. He woke up the next morning as the god of bravery, freedom, and alcohol. He still doesn't remember how he did it.
There's also the infamous OldManHenderson, the only man who ever won at Call of Cthulhu. Specifically, he did this by BLOWING UP GOD-DAMN HASTUR WITH AN ICE RINK FILLED WITH DYNAMITE AND C-4.
The crazy part comes in when you realize that this is a heavily MinMaxed Schizophrenic Scotsman who's convinced he fought in Vietnam when he really didn't (he's too young), and is a Crazy Survivalist to boot. He also joined the investigators solely because he thought Hastur had stolen his lawn gnomes.
Hearing the story, his player certainly qualifies, by writing up a 320-page character background before play started to justify everything he ended up doing in the ensuing game sessionsnote or so the DM thought; it was actually about 100 pages long, and the player would change certain details over time.
Including, purportedly, an entire section written in grammatically perfect conversational German, and Henderson himself being perfectly fluent in Portuguese; the player was fluent in neither at the time. The words "Fell Mood" come to mind - this guy did NOT like being railroaded, and made his GM regret everything.
Namely: The fellow who created Henderson was described as a very cool and mellow guy until you started intentionally jerking his chain. Henderson came about because the GM was busy causing TPKs whenever and wherever he could just because. In the creator's words: "This guy was a dick who had it coming." He didn't mind being railroaded, but the way that particular GM was doing it was what set the guy off.
Mutant Chronicles: The Order of the Monkey, House Dante's house troops, are considered this in-universe. In peace-time, everyone thinks they're a bunch of useless, drunken Frat Bros in armor who play stupid pranks at the least convenient times, and that a smiling monkey is a pretty stupid crest. In war, everyone is terrified by the booby-traps, unpredictability and terror tactics the Order uses, and people remember that monkeys smile to show aggression, not happiness.