Nextwave's Aaron Stack. "My robot brain needs beer."
Honestly, everyone in Nextwave, which is the point.
Me (Deadpool) being crazy awesome was key in defeating the Taskmaster in hand-to-hand combat. I don't mean to brag, but that's no small feat, considering that TM's powers allow him to copy and second-guess anyone's fighting style, including, but not limited to, Captain America, Daredevil, and any other fighter in the entire Marvel Universe. However, I am so crazy sexy, I don't need a recognizable fighting style. My brain having exploded from my sheer, goddamn handsomeness also makes me near-impossible to mind-read or mind-control, and let it be known that I was once (fyi, this "once" I'm talking about started when Joe Kelly adopted me, and it's still going) the only person who could save humanity by a combination of this and my willingness to kick Captain America inna nerts to do so.
For the record, I defeated Taskmaster by just being myself. Instead of attacking Taskmaster directly, as he was obviously expecting, I broke into a dance routine. And not just any dance, mind you, but the Macarena!
Uuh, and then I dodged a RPG launched by Bullseye, the Man Who Never Misses, by turning my truck around and rolling down the window. Even Bullseye admitted that it was, like everything else I do, pretty awesome.
As you might have guessed, the fans like me because, in spite of my sheer sexiness, I'm still willing to talk to you unlike all the other protagonists.
Bullseye sometimes. Especially when he's really creative with his ability to use anything as a weapon.
This is a man who, after being paralyzed and captured, had to be put on stool softeners and a liquid diet so that he couldn't kill his guards with his own waste. And he even says that he could probably pull it off!
Rorschach of Watchmen generally switched between crazy and awesome, but occasionally both at the same time, such as when he ambushed a guy by hiding in his fridge, and ambushed him again by hiding near the fridge and jumping out when the guy reads the note in the fridge saying to turn around.
"No. You do not understand. None of you understand. I am not locked up in here with you. You are locked up in here with me."
The first examples might only seem crazy since he was ambushing an innocent man. But it was a full grown man inside of a fridge for what we can assume was possibly hours just waiting. Crazy. Awesome.
Transmetropolitan's protagonist Spider Jerusalem. After beating the snot out of a thinly-veiled grown-up Charlie Brown to get to a source of information, he throws another of her bodyguards out a barroom window... and onto a dog who is a dead ringer for Snoopy. (This being Transmetropolitan, it might even be an intelligent dog.) Whose corpse he claims as his dinner.
Snowflame from New Guardians #2 is a supervillain whose powers come from snorting massive amounts of cocaine, worships cocaine, and is basically the religious leader of a cult that also worships cocaine.
Green Lantern: Larfleeze, the only member of the Orange Corps of Greed. He had an I'm Taking Her Home with Me! moment with Scar, a corrupted, demonic Guardian who serves Nekron because.... she was evil, so she was different than other Guardians, so she was worth more. And he really wanted his own Guardian.
There's also the fact that his defining character trait, greed, leeds to his power being more effective. Unlike the lights of the other Corps, the Orange light of greed is best when wielded by only one person.
Then there's Dex-Starr, an extremely dangerous member of the Red Lantern Corps....who also happens to be a cat. (His origin is almost Tear Jerker, too.)
Ben Templesmith's (of 30 Days of Night fame) Welcome to Hoxford, about a private prison run by werewolves, brings us delusional serial killer Ray Delgado, whose first reaction to a werewolf snarling in his face is to bite off its tongue. Of course, he did believe himself to be Chronos, Lord of the Titans at the time. He later decides he's a werewolf too, challenges the ten foot tall alpha, and successfully chews out its throat.
Mike Allred's Madman is, well, a madman. In the first few pages of his first appearance, he knocks a guy out with a lead-filled yo-yo before prying one of his eyes out and eating it. He was doing this to psych out another malcontent, you see. ""Now... be quiet. I'm going to touch you." To be fair to Madman, he threw the eye up when the coast was clear. The point is that he's crazy enough to try anything to get what he wants.
Although this is somewhat of a case of Early-Installment Weirdness. Allred didn't even like that scene right after he wrote it. For the rest of the series, Madman was much more childlike.
Rex The Wonder Dog is made of this. Initially, he was simply an ordinary, non-talking non-magical dog that could fish using a rod and reel, drive cars and boats, had a successful career as a newspaper photographer, and once killed a T. rex using a nuclear bomb. Then it was revealed that he was an decorated war hero from World War II, and a Super Soldier with a similar origin to Captain America. Then, after all the newspaper photography career and T. rex killing, he drank from the fountain of youth, gaining the ability to speak the language of every thing that lives on planet Earth, as well as eternal youth and unspecified magical powers noted to be among the strongest on Earth.
Practically every protagonist in Sin City. They may be violent and crazy but they're also fun as hell.
Lex Luthor, especially in the Silver Age. He once cured cancer just to trick Superman into believing he'd reformed. Plus he's a normal human who is the greatest enemy of one of the most powerful superheroes ever created.
Issue 23 of IDW's Transformers Ongoing gives us crazy awesome Orion Pax (Optimus Prime before he had the Matrix), a tough no-nonsense police captain who, when his subordinates are murdered by senate goons trying to free Whirl (a bullying officer under Prime who himself was jailed for beating on a wrongfully-incarcerated pre-evil Megatron), ends up having to use the corpse of his lieutenant Springarm in motorcycle mode to dispatch of the two goons... one of them by tearing one of his smokestacks off his shoulder and stabbing it in said goon's face.
And speaking of Whirl, Whirl. It's mentioned in More Than Meets The Eye that he punched himself in the fact to prove that he's super invincible.
Moon Knight , especially when they play up the multiple personalities.
On the very literal end of things: Crazy Jane of the Doom Patrol.
30 Days of Night brings us Lex Nova, the deranged vampiric detective. He monologues out loud to himself without knowing it, and seems to truly believe that he's in some kind of insane detective novel. He still somehow manages to be effective, and really, one of the nicest vampires in the series.
The Boys has The Frenchman, who might not be legally insane, but certainly acts like he is. While he has his moments of lucidity, and he certainly displays competence in the more cerebral parts of The Boys' line of work, he also often goes into non-sequiturs, is friends with The Female, and his main contribution to the team is as muscle, AKA putting on his goggles and going apeshit, despite his otherwise genial and mild-mannered nature. While the origins of each other member of The Boys, past and present, have been told during the series, The Frenchman's past as he tells it is not even remotely believable, and the other characters clearly state at the end of his story that he might not be entirely there.
Mike Baron's The Badger is entirely about this trope... he would be absolutely nothing without his multiple personality disorder, and its associated powers. One enemy even said it straight out: "You are WORSE than sorcerous! You are completely insane!"
Scud The Disposable Assassin is a comic series that runs on Crazy Awesome. The first issue has the titular Scud, a robot assassin bought out of a vending machine, fighting a monster with mousetraps for hands, a three-pronged plug for a head, and mouths on it's knees. And it only gets weirder from there, including a cult that worships manliness and unnecessary explosions, an immortal voodoo priest Benjamin Franklin controlling zombie dinosaurs, and a werewolf astronaut looking to gain power from standing on the moon.