Lydia Bennet from The Lizzie Bennet Diaries. When she was ten, she made her father promise that if the world's largest energy company fell to bankruptcy, she would have a pony. After Enron sadly fell to bankruptcy, she got a pony. When he left, she got a cat. She named her Kitty, and gave her a twitter account. Now, they both have a twitter account on which Lydia socializes and influences her admirers. Using her social life and wits, she made herself invited to a party with Mr. Bing Lee. Since her sister has a crush on him, she made sure he would promise to host a party at his home, waiting until he was drunk. After the party, he and her sister spent the whole evening talking to one another, and her sister now lives at his place (episode 28).
Reginald Cuftbert's MO is this, in the room with two people you need to kill? Blow up the room, while still in it! Crazy monsters walking a casino? Punch them out! Want revenge on someone? Blow up his pants, stealthily! Annoyed? commit genocide! or alternatively punch the people annoying you out of existence by accident.
In the League of Intergalactic Cosmic Champions Mr. Absurd, once when escaping through a portable hole he reached back through the hole, grabbed the edge & pulled the hole through the hole so they couldn't be followed.
Most, if not all things, by John Su, but most recently this.
Everything Doctor Steel does is cool, awesome, and/or funny, over-the-top, or a combination thereof.
Assignment 2's Stanley Brown oh, so much. Swordfights with genetically-enhanced monkeys are pretty much par for the course. In a world which is almost logical and normal, he basically does whatever the hell he wants, ignoring anything remotely resembling sense.
Jade Sinclair, codename Generator, of the Whateley Universe. What has she not done? She has defeated opponents with her flying Hello Kitty makeup compact. She once defeated a guy in power armor who was holding her hostage by impaling herself on his knife. She took over part of a Syndicate hardsite by making everyone think she had turned dead thugs into zombies. She let a Mad Scientist experiment on her, including he cut her foot off! And she let him do it. She got better. When she shouted, "People! I am not crazy! And the voice in my head agrees with me!" her friends laughed, because it's exactly true. Is there something more crazy and awesome than just Crazy Awesome?
Don't forget her shoulder angel thing, a prank that went horribly right, nearly launching the entire campus into a warzone; and the time she got to baby-sit for an Expy of the Fantastic Four. Oh, and her cabbit of doom.
What about the time she rescued Fey by shooting an RPG at her? What about the fact that Whateley Academy has special armbands for the school Ultraviolents, and special armbands for the school pacifists, and she has one of each? What about the time she defeated Solange by making Solange shave her own head? What about the time she drove her own team crazy with an animated cabbit? What about the fact that she turned that animated cabbit into a Killer Rabbit that among other things did a Groin Attack out of your nightmares?
Or her latest idea: a radioactive condor suit for Tennyo. The simulation instructors are worried, and they haven't even seen it yet.
Even more bizarre, the Radioactive Condor Girl plan actually worked, against a simulation that no team at Whateley Academy has ever beaten on the first try, and few have ever beaten, period. Right now, it's still a Noodle Incident. Maybe an Offscreen Moment of Awesome. And they're forbidden from doing that ever again.
Dr. Bright too. He once tried to disprove the existence of 343 (who's basically God), to 343.
The SPC, or "Shark Punching Center". They have special procedures as to exactly how and when each species of shark needs to be punched.
Just about everyone in Red vs. Blue is this at some point or another.
Tobuscus has shown signs of this. The most obvious example is when he took contact with an auction website about them giving him an Ipad because he had spent about 500 dollars on their website:
Tobuscus: (...)I bet on an Ipad like 500 times and lost, can you just retroactively give me one for free?
Tobuscus: Okay.(...) I have an idea. How about you give me two, and i give one away on my YouTube channel?
Spokesperson: That's not a compromise, you just added another iPad!
Tobuscus: Shhh. Let me finish. Also give anyone who signs up for your website through my channel five dollars. Deal?
Spokesperson: WHAT! You just added another thing, are you nuts?
Spokesperson: Oh. Alright, let's do it.
Evan from Everyman HYBRID tried to clobber Slender Man with a baseball bat. It didn't work, but... well, he attacked SlenderMan with a baseball bat. And lived. Later he manages to not only survive a close encounter with the Rake, he than manages to throw it out a second story window!
Speaking of the Mythos, Maduin pretty much has this as his schtick. Among other things, he once marched straight up to Slendy, handed him a twenty-dollar bill, then just turned and walked away, leaving the Eldritch Abomination standing on the sidewalk and presumably wondering what the hell just happened.
Up to Eleven when Maduin puts a face mask on TPF and he goes bananas. Earlier he caught a proxy, knocked it out, tied it up, set it in front of a mirror and had it dressed like Slendy. Maduin is the king of Crazy Awesome.
In Trinton Chronicles Peter is a good example of this as his powers are over dark matter and black hole creation.
This animation by Neil Cicierega, AKA Lemon Demon. Yes, the Lemon Demon responsible the Ultimate Showdown of Ultimate Destiny. Which also qualifies. In fact, just about everything this guy has made qualifies.
The parts of Fine Structure that aren't made of nightmares are made entirely of this.
Oul is due to arrive at Earth in a few hours, at which point it's game over. The final solution to defeat Oul will take 20,000 years to calculate, give or take a few millenia. John Zhang builds an information-to-energy converter in his thoughts, feeds the Script in one end so it will generate more energy than the entire universe can ever possibly generate, builds an Albicurrie drive in his Klick device, and takes off for the other end of the universe much faster than the speed of light, and accelerating logarithmically - just to force the Imprisoning God to wall off the solar system from the rest of the universe to contain humanity's reality-breaking Crazy Awesome and buy the necessary 20,000 years to figure out how to beat Oul.
Ching tries to destroy Xio in the modern day, forcing Xio to throw him forwards in time. He lands in the future just as the Solution is finished, beats Xio to it, earths all of Xio's Power in himself, and takes on Oul one-on-one, killing it with a bullet made from the entire core of the Earth. In the ensuing few seconds of existence, humans from alternate realities swoop in and save everyone still alive and most of the historically important artifacts and structures, and spends the remaining milliseconds before Earth is destroyed watching the spectacle.
Shadowhunter Peril's Veronica Marie Carter falls into this category easily, because she is able to act like a total lunatic yet hold her own in a fight against multiple enemies. She tends to border on the bipolar side occasionally, switching from maternal figure to warrior princess to abusive friend in a matter of moments. That, and the fact that she also blows shit up for absolutely no reason whatsoever.
Fegelein in Hitler Rants. Not even the laws of physics will stop him from committing increasingly surreal and hilarious antics against Hitler. The other Masters of Antics also qualify, but not to the same extent.
Nora Valkyrie from RWBY. Outside of battle, she comes up with odd plans, like using sloth calls to send a secret message. Despite being told that sloths don't make a lot of noise, she deems that this makes this plan even more perfect. In battle, she uses her Magnhild, a rocket-powered warhammer capable of converting into an M32 grenade launcher, with wild abandon. Using Magnhild, she can actually use the rocket power as a Recoil Boost, and generally flies through the air while giggling like a maniac. Nora is one part Cloud Cuckoolander, one part fearless warrior, one part incorrigibly cheery, and one part Blood Knight, all wrapped up in an adorable but psycho shell.
Angry Joe helping Linkara defeat Lord Vyce with an army of Angry Joes in the Doctor Who Classics 17 review on Atop the Fourth Wall.
Parody Janeway from SF Debris, a delusional, unstable, ruthless petty tyrant and would-be Mad Scientist who relies on advice from her shoulder spider, orders her crew (okay, just Harry) to be footstools, and is capable of succeeding through lunacy at things no sane person would even attempt. At one point, her master plan is to hijack the Borg Collective by being assimilated. And it works.
Psycho Gecko, the main character from World Domination In Retrospect, may be evil, but he's evil in a Crazy Awesome way that makes it difficult for superheroes to fight him effectively. He hides lasers in potato peelers and makes rubber chicken grenades. He's not good at throwing knives, so he added explosives to them. He once brought a pineapple along on an attack because he wanted to know what it felt like to hit someone with a pineapple. By the end, he's used it to take down a helicopter.
Saige from Deadly Space Action has an assortment of weird hobbies and fixations, but is probably the most consistently competent member of the crew.
Gordon Freeman of Freeman's Mind. Ross Scott (the creator) said that he went for a mentally unhinged and quirky characterization because he couldn't see a normal person surviving the events of Half-Life. His more eccentric attributes do indeed save his life a few times. In later episodes, he also gets pretty lax with his personal safety, seeing as how he considers running right past a Bradley through an open killing field while it attempts to hit him with its autocannon to be a completely viable strategy.