Adrian Monk is arguably the paradigm for this type. His obsessive/compulsive disorder got him booted from the SFPD and makes it difficult for him to get through his day-to-day life, but it also aids him immeasurably in solving cases. As Monk himself often acknowledges, "It's a gift... and a curse." There once was an episode in which Monk was on pills which took away the Crazy, making him also lose the Awesome.
While most of the cast technically qualifies, the Australian TV show Double The Fist gets its crazy on in the form of Mephisto/James Wenham. Example: In one episode he was stranded on a mountain. So, remembering his security guard training, he found a dead snowboarder and rode him down the mountain. He later uses the body as a melee weapon. DAYUM!
The Doctor. I mean, this is a guy who saves the universe using a Time Machine shaped like a police box and a screwdriver.
Sarah Jane: You're serious, aren't you? The Doctor: About what I do, yes. Not necessarily the way I do it.
Season 5/31 is filled with this thanks to the Eleventh Doctor and Amy Pond.
Amy Pond throws a tantrum at her wedding that pits her memory against the very fabric of reality. The fabric of reality promptly throws up its hands, squeaks "uncle!", and allows the Doctor to exist again, because she's the type of girl who bit her psychiatrists when they tried to convince her that her imaginary friend wasn't real, and stubbornly refuses to listen to the entire universe telling her that there's no such thing as stars all her life, because she remembers them, dammit, and to hell with anyone who dares imply she's remembering wrongly.
Eleven is the epitome of Crazy Awesome. Who else would eat fish custard? Who else would jump in front of a gun and shout "Look at me, I'm a target!" complete with a manic grin and a thumbs up!? And, to put it bluntly, who else would look awesome saving the universe in a fez and a bowtie while carrying a mop?
The Eleventh Doctor: I am definitely a madman with a box.
This is merely scratching the surface of Eleven's oddness. Mention also needs to go to biting Amy, psychic headbutts, bluffing Daleks with a cookie, bursting out of a stripper cake, and getting a Weeping Angel to say "comfy chair."
He also blows up an entire Cyberfleet, just to make a point!
While 11 is definitely one of the most alien Doctors, and 4 at a close second, they all have it. There's 5 with the celery on his lapel, 9 who swaps guns for bananas and 10 who defeats the Cybermen and Daleks with 3-D glasses.
To go deeper; Four wore a ridiculously long scarf and greeted everyone, even those trying to kill him, with a calm "would you like a jelly baby?" Six sported a ridiculous rainbow coat and honestly couldn't care what anybody thought about it. In Eight's only televised adventure he went on raving about his shoes and aiming a gun at his own head. Ten, in addition to the 3-D glasses, defeated his first enemy with a satsuma....should we go on?
I would like to take this opportunity to point out that the First Doctor's original plan to help the Greeks defeat the Trojans was to build giant paper airplanes and catapult them over the city walls. He ended up scrapping that idea and suggesting a giant wooden horse instead.....
You forgot the part where he's dancing while doing all of the above. Badly.
You also missed the part where he puts on a gas mask and promptly gases his whole cabinet, killing them, then answers the statement, "YOU'RE INSANE!" with a thumbs up!
The original Masters were Crazy Awesome, too! In his first story, the Delgado Master killed a man by feeding him to a chair and tried to take over the world with plastic daffodils. In his first story, the Ainley Master destroyed one tenth of the universe by accident.
And then using the incident as leverage to bluff the Universe into submission. And that's just scratching the surface.
On the subject of the old series, how about Mad Bombernote Which, incidentally, makes her the only Companion until Amy with canonical mental illness Companion Ace, who actually inspired the creation of the Crowning Moment of Awesome page by beating the crap out of a Dalek with a baseball bat?
Mr. Bean. Seriously, dressing while driving (with his feet!) a car, foiling a car thief by taking his steering wheel with him to a picnic, winning a dog show with a teddy bear? And that just a couple of episodes...
Cannot forget the episode where he tied a couch above his car (a Mini) and sit on it because he didn't want to pay delivery.
He paints his apartment with a can of paint and an explosive. He can do it even when the MythBusters say it doesn't work.
He also manages to headbutt the queen into unconsciousness and get away. Not that it was intentional.
Charlie Crews, the ex-con police detective in Life. Justified in that his settlement with the LAPD for being framed and imprisoned for twelve years before being proved innocent included (in addition to substantial financial recompense) the restoration of his position on the LAPD and his promotion to detective (he was a patrolman when he was framed and jailed)... for as long as he wants it. He's got very expensive lawyers keeping him on the job, no matter how Crazy his Awesome might get.
Hooch from Scrubs is one of the more popular minor characters due to him being very Axe Crazy.
Hooch: Who the hell... put bouillon cubes in the shower head!?! Huh? Hm, did you do it? Hm? Did you? If it happens again, I will wait in my S.U.V., blast me some speed-metal—5.1 surround sound, heavy on the bass—and someone... will be getting...mowed...down.
The Janitor also qualifies as this. He demonstrates incredible skills throughout the series, all in ridiculous scenarios.
Gene Hunt. Drinks on the job, abuses his colleagues, insults everyone in sight, has regular punch ups with his DI. And always gets the baddie.
Jeremy Clarkson of Top Gear. In the show's various challenges (convert a vehicle to be amphibious and drive it across a body of water; buy a cheap two-wheel-drive car and cross Botswana), he inevitably goes for more flash, more speed, and more power. As a result, he always seems to achieve magnificent success or equally magnificent Epic Failure.
Really, all of them. James May actually drank a Bovril, beef and brick smoothie; the Hamster is in love with a 1963 Opel Kadett and once started eating a photo of a car to prevent it going on the Cool Wall; and Clarkson built a V8 rocking chair and took a chainsaw to a photo of a motorbike.
Amerca's Funniest Home Videos will have these kinds of people from time to time send in a video. Like the guy who used an American Civil War era cannon to blow up a snowman.
Q, from Star Trek: The Next Generation has the Magnificent Bastard thing down to a tee, and he certainly has his moments of crazy awesome throughout the series run, too. (Instigating the end of the universe via Picard due to orders by the Q-Continuum and then manipulating Picard into saving it? Totally counts as awesome).
The Mariachi band is all one needs to see to understand how Crazy Awesome Q is. Special mention also to the Robin Hood episode.
Les Nesman from both WKRP in Cincinnati and its Revival. He has an "office"—actually just part of the office floor. But "one day" he will get walls, and he has put down tape strips to mark the future walls and door. He even insists that people knock before entering. Les feels so strongly about this that in the first series, when the only woman on Earth who would sleep with him decides to remove the tape, he kicks her out of his life. Here's the kicker: it's catching. In an episode of the second series, Les has vital information in his desk, but he has "locked" the "door" to his "office" and taken the key. With Les not even in the room, the other characters admit that the "locked door" is an obstacle, but one of them tries to "jimmy" the "lock" with a credit card. And he fails.
On the other side of the coin is Jubal Early, who manages to combine crazed, philosophical rambling with a keen understanding of psychology to manipulate the entire crew while seeming utterly insane.
And, rather fittingly, River ends up blowing him out of the water at his own game.
Kramer from Seinfeld somehow manages to be the most successful character in the series despite not doing anything approaching reasonable. When he advises others to do things the way he would, they inevitably fail to inject enough Awesome into their Crazy.
Heroes: Sylar is a completely insane serial killer, but he's so nonchalant about everything and can be quite funny when the writers aren't filling him with Wangst. He's the only character who actually enjoys his powers and he's hilarious when he takes on a fake identity (usually just for fun).
What about Hiro Nakamura, the self-proclaimed "Masutah of Tahm ond Spehce"? The one character who takes Genre Savvy to new levels and wanted to be a hero from the start, who manages to break Suresh out of a psychiatric hospital while his brain is scrambled, among other things. The only reason Sylar hasn't been defeated by him is that Hiro believes he has to let Sylar exist. Samuel is a threat is due to having a hostage. Not to mention he committed one of the most Nightmare Fuel-riffic vengeances in TV history. The writers had to nerf his powers to keep him from one-shotting all comers.
Dr. K from Power Rangers RPM, who has twice taken out The Dragon using a violin, and is so forced and unnatural in social situations that it goes past awkward, past being simply funny, and into pure awesome. One of the two violinnings was with an amplifier to create a sonic weapon, and once was by using the violin to remotely control the base's ventilation system. She also keeps a laser cannon in the refrigerator (ready to fire the moment the door is opened), has created an experimental clothes-destroying weapon, and once blackmailed the person who is ostensibly her superior with the threat of her maybe kinda possibly having worked on an undetectable chemical that causes a case of diarrhea "a thousand times more extreme than the worst case of dysentery ever recorded." If it was anyone else, it would sound ridiculous and be seen as a harmless joke. With her, he caved instantly.
Speaking of which, then there's Gem and Gemma. Sure, they're called the MarmiteTwins, but they emphasize the CRAZY part of this trope. (Boom time, anyone?) Although completely childish, drawing unicorns and rainbows as maps, and completely creepy when they engage in twin-speak, they are highly intelligent and often come to the other Rangers' rescue. They're even the catalyst for Dr. K's social development as they're the only two friends she had growing up together in Alphabet Soup, and they call her out for being impersonal towards the other Series Operators when they turn up alive from the wastelands.
Duct tape boat. Waterskiing behind a cruise ship. Dropping a car from 4000 feet. Making a seesaw that can take something like 40000 pounds of force without breaking or even bending and still works as a seesaw. (Might I add that it is pink?)
Crossing a hundred yard wide drydock in a bridge they made out of duct tape, while being afraid of heights? .
They made not one but two hovercrafts.
Four functioning historical weapons: The confederate rocket, the confederate steam machine gun, the greek machine-bow and the Korean HWACHA!
You can't make a sailboat sail by blowing on the sail with a fan that's mounted in the sailboat itself. It's against the laws of physics. They built one. And made it work. It worked. It actually sailed. Do you understand what I'm saying? They broke a law of physics. "I reject your reality and substitute my own" is all that needs to be said.
Finally, anything that follows the phrase "Replicate the result."
This is pretty much Jack O'Neill's job on Stargate SG-1, given that he's essentially an avatar for the Tau'ri (Earth humans) at large. In the SG-1 mileu, Puny Earthlings aren't so puny, by virtue of overturning every single conception of How Things Are Supposed To Be by the older races. Note that the Tok'ra had been fighting the Goa'uld for millennia with little progress. But the SGC, with comparatively primitive weapons and tech, beat them in 8 years. Crazy Awesome is Earth's hat!
In Dollhouse, Alpha spends most of his screen time being evil and ridiculously awesome.
Renee Walker in season 8 of 24, after going through a season-long Break the Cutie the previous year. She came out the other side as a Death Seeker willing to dismember a man if it means not blowing the mission.
"I don't know where this is headed, but the only one of us with the coordinates for this destination in his hardware is you. Go on whatever vision quest you require. Stand on the rim of the volcano, stand alone and do your dance. Just find this beast before he takes another bite."
While Tyler in My Hero is usually a Cloudcuckoolander, he jumped into this trope briefly when George was having trouble concentrating and brought home an armed missile. Tyler promptly disarmed it with a code he got off the Internet. This qualifies because:
He had no idea that there was even going to be a missile, which would make him Crazy-Prepared if...
...he had ever prepared for anything previously.
House. Once an Episode he thwarts proper medical procedures, medical ethics, or common sense. And pretty much every episode he fixes a patient no other doctor could.
Correction: he diagnoses a patient no other doctor could. Sometimes, his diagnosis reveals an incurable condition, and most of the time, the actual therapeutic procedures are left to other doctors.
In one arc he starts using methadone instead of his usual hydrocodone. It works very well, eliminating his leg pain (rather than simply reducing it as the Vicodin does). Without the pain, he mellows and becomes less crazy... and less awesome. Eventually he decides he likes being awesome more than he likes being pain-free.
Winston: Did you just flip a coin? Guerrero: ... No.
In the second season premiere, he walks into a fancy cocktail party unannounced, wearing a tuxedo and casually carrying a sniper rifle slung over his shoulder. He's genuinely surprised that the security overreacts.
Showa-Era Kamen Rider, being the franchise that gave us such things as Wizard Starfish Hitler, has a fair number of such characters, but special mention must be given to Kamen Rider Stronger. Stronger is THE WARRIOR OF JUSTICE. Where other cyborg riders were unwilling victims rescued at the last moment before brainwashing, Stronger deliberately underwent the procedure and beat the brainwashing through pure force of will, at which point he beat the shit out of his captors. Stronger's Super-Mode is so hardcore that he must exhaust its power when using it, or explode. Stronger can punch a Kaijin so hard that it will not only be reduced to a twinkle in the sky, but will explode shortly thereafter. Stronger once beat the shit out of Heath Ledger's Time-Displaced Ghost, at which point he exploded. Guns are sexually attracted to Stronger, as shown by the opening title sequence.
The Owner of the Den-Liner. To specify, he has a habit of eating plates of food while trying to avoid tipping over a small flag in the center. Crazy? Yes. Having a gun that shoots a barrage of these flags as sleeping darts used to knock out an entire room full of people in seconds? Using one of these flags to take down a forcefield? Crazy Awesome. Running along side a time traveling bullet train that's going at full speed? Using a normal everyday bicycle to POWER said time traveling bullet train with his own leg power? Crazy AwesomeUp to Eleven!
Continuing our Kamen Rider tradition is Gentaro Kisaragi. A delinquent who has one mission: make friends with everyone in his new school. His first moment of awesome? Screaming at a fellow student for dumping a girl's love letter to him and then jumping into a cold river to pick it up. Then he has crazy weapons for his arms and legs like a paintbrush, faucet, drill, rocket, etc. Oh and his screams are enough to be heard across the universe and back, not to mention that he has pompadour sense.
Lowell Mather (Thomas Haden Church's character) Tyler in Wings whose sole reason for existence in that show was to alternate Stupid Lowell Tricks with Crazy Awesome stunts.
Occasional episodes/dialogs with Joey of Friends fame
The A-Team: There's a good reason why he's called "Howlin' Mad" Murdock. He once created a successful diversion by impersonating a horse, and in the episode after that, when Hannibal's plan failed, Murdock managed to fix it just in time. And he has an invisible dog named Billy.
Not to mention the time that he insisted he could become invisible at will—the subject of much joking among the Team, who could see him quite clearly—and then proceeded to do precisely that to infiltrate a restaurant in search of a suspect.
This is pretty much the definition of Parker on Leverage.
Her solution to get Sophie over twenty floors down to the bottom floor of a skyscraper? Clip herself and Sophie to a vertical zipline that she designed herself that runs down the middle of the staircase and go, after forcibly dragging Sophie over the rail. And it hadn't been tested. Ever. (Or at least, if it was tested that time there was a similar incident where it wasn't.)
John Crichton of Farscape becomes like this as the series progresses.
One particularly awesome example is dancing on a table in front of Scarran Emperor Staleek with a NUCLEAR BOMB strapped to his hip. One of Staleek's attendants call Crichton insane, to which Aeryn responds, while grinning and having eyes only for her nutty boyfriend, "Isn't it fun?"
America's Got Talent, of all shows, brings us ArcAttack, who have pioneered the use of Tesla coils as a musical instrument. For real. And it gets better... The founder of the group is one of the group of guys who invented the singing Tesla coil in the first place. And their act is so dangerous and uses so much power that after the audition, they weren't allowed to perform in the theater, and had to have a special stage built for them outside.
Morgan Grimes from Chuck, after he joins Team Bartowski, does such things use himself as live bait to get the Villain of the Week's bengal tiger into Ellie and Awesome's empty apartment and grab a hanging live wire and swing down into a pool of water to rescue the rest of the team.
How is Barney Stinson from How I Met Your Mother not on this list? Where his shame gland should be there's a second awesome gland!