We Do the Impossible

"The difficult we do at once. The impossible takes a little longer."

—Unofficial US Army Corps of Engineers motto

The character hasn't just beaten the odds once. He's survived several Suicide Missions. The Uriah Gambit failed. He won the Last Stand. In short, the hero has repeatedly accomplished so-called Impossible Tasks. And his superiors or the world at large have taken notice... and unfortunately for the poor hero, he or she is now the go-to person for missions that are thought impossible for anyone else.

Characters like this can get this reputation a number of ways.

Surprisingly Elite Cannon Fodder: The hero or heroes survive their superiors trying to kill them so many times deliberately in less important missions that when a crucial task that seems impossible actually does come up, they are the go-to group.
The character really is that Badass. They've overcome so many situations that others would have died in that they have accidentally built up a reputation for surviving anything. Nothing deliberate on the superiors part, but the odds came up and they beat them.
Training from Hell: The characters might not have actually beaten anything yet, but they were put through training that by all rights shouldn't be too much easier than the supposed impossible mission.
It's entirely undeserved or grossly blown out of proportion. Rumor and exaggeration may turn a simple feat into an epic accomplishment, or make them up wholecloth. This may or may not be because The Powers That Be enjoy seeing the character win and/or suffer.

One possible deconstruction of this trope is that the Big Good, Big Bad, or some other powerful or manipulative figure has been arranging things such that the characters always succeed, or at least have even odds. Once this aid is exposed, they have to face real impossible missions.

Supertrope to Surprisingly Elite Cannon Fodder. One of the reasons It's Up To the Player Character, One-Man Army, and One Riot, One Ranger. In Video Games, Save Scumming probably helps. Compare Impossible Thief, who does (usually even more blatantly) impossible things in a different context.

Examples of We Do the Impossible include:

Anime and Manga

Tengen Toppa Gurren Lagann gives us Kamina and his motto : "Go beyond the impossible and kick reason to the curb !" The trick : in a universe that runs on Hot-Blooded-ness, it works. And it is contagious, as Simon and the whole Gurren Brigade demonstrate.
This is essentially the overall main plot of The Irresponsible Captain Tylor, both the titular character and the crew of the Soyokaze are either type 1 or type 2, or some combination of both (the series doesn't make it clear which).
Legend of the Galactic Heroes: Yang Wen-Li is nicknamed "The Magician" for his ability to emerge victorious against ridiculous odds: the first time he captured Iselhorn fortress, it was in fact a Suicide Mission given by superiors who wanted to get rid of him: he emerged successful from said suicide mission with insulting ease
Shirou Emiya in Fate/stay night survives battles against Servants where other Servants have failed. Type 3, since it's revealed that his magical self-training is physically demanding to the point of life-threatening, and has conditioned him to be more powerful and durable than Rin's more traditional study.
- His father, Kiritsugu Emiya, in Fate/Zero is hired for his reputation as an effective assassin against even the most powerful mages, mostly for fighting dirty.
Trigun: Vash the Stampede is a type 2.
Mu La Flaga in Mobile Suit Gundam SEED uses "I am a man who can make the impossible possible," as his Catch Phrase. He lives up to it with a combination of Ace Pilot skill, ingenuity, a healthy dose of common sense born of previous experience, and a bit of pure luck, to the point that in the penultimate episode he blocks an antimatter cannon with his Humongous Mecha and succeeds in shielding the ship behind him even though by all rights the positron beam should have gone right through his mobile suit and taken the ship out anyhow. And then the sequel reveals that he lived through it.
Issei Hyodou, the protagonist of High School DxD, can be considered as this. Seriously, who can pull off crashing an engagement party and declare that Rias' virginity belongs to him, take his rival's power and infuse it with him even though said rival's power is the total opposite of his, and summon the freaking god of breasts that makes even gods from other mythologies go "wtf".

Comic Books

Damage Control does this for construction.

The City of New York tried to fix the George Washington bridge for 7 months. Then they called us.
We fixed it in one day. Before lunch.

Film

The Fifth Element: Korben Dallas is types 2 and 3. Helps that he's Bruce Willis, which crops up in many of his roles.
Die Hard: Never mind anything else this overworked NYPD cop does, he takes out a helicopter with a police car. Types 1 and 2 (though not intentional canon fodder, pretty much hardly ever, he's just a mook that always ends up in the wrong place at the right time, or as they say in Live Free Or Die Hard: "That's what makes you That Guy."
John Rambo arguably meets all three criteria.
As does Riddick.
In The League of Extraordinary Gentlemen, Alan Quartermain (Sean Connery) apparently actually hired a guy to screen visitors because of his reputation for (2).
Mission: Impossible teams seem to fill this role.

Literature

The title character of The Wonderful Wizard of Oz is a subversion. He's got a rep for being the great and powerful wizard who can solve any problem, but he's really a fake.
Rogue Squadron of the X Wing Series, and its commander Wedge Antilles in particular. One of the Rogues' mottoes is "Impossible is our stock in trade, and success is what we deliver." No matter how many Rogues die in combat, Wedge always survives, and the squadron always rebuilds.
- In a book of the New Jedi Order, Wedge isn't one of the Rogues anymore, instead commanding a good-sized portion of the New Republic's fleet in the days after Coruscant fell to the Vong. What's left of the New Republic's ruling council tells him to hold Borleias as long as possible. Wedge is highly savvy and sees through them, aware that they want him and his entire command to get gloriously killed so that they can surrender later. He bullies them into giving him more resources by threatening to quit, along with most of his command staff, if they don't. Ultimately he sets up a new Rebel Alliance to work quietly under the radar, and while he ultimately retreats away from Borleias, he gives the Vong one hell of a bloody nose on the way out.
- In the same duology of books, after setting up their base on Borleais, an enemy fleet attempts to take back a temporary base they're fabricating on the moon. Wedge and Tycho very specifically plan this operation so as to make it look like they lose the moon and run back to the planet in retreat. Due to coincidental timing by the cavalry and their Big Damn Gunship, they end up destroying the entire fleet instead.

Tycho: "We'll put that in your biography. Wedge Antilles was so great he couldn't lose when he tried."

Ciaphas Cain's stock-in-trade, much to his annoyance. A mixture of types 4 and 2, odd as that is. He typically gets sent in to do something "merely" difficult, tries to take the easy way out, finds out things are much worse than they seem, and ends up doing the impossible anyway, only further enhancing his reputation.
- Just one example: In Death or Glory, he's shot down onto a strange planet in the middle of a continent controlled by Orkish invaders. His plan: get the frak out of there as soon as possible. The result: becoming a rallying point for a growing army of survivors, destroying thousands of Orks, and personally duelling the Ork Warboss to (the Ork's) death, effecting the liberation of the planet almost personally.
Vorkosigan Saga: The Vorkosigan Family. As Cordelia remarks; "The difficult we do at once. The impossible takes a little longer."
James Bond. Even more so in the films.
Some Discworld characters. Lu Tze is types 2 and 4 simultaneously. Granny Weatherwax is a type 2.
Mat Cauthon, Perrin Aybara, Lan, Thom Merril, even Jain Farstrider from Wheel of Time have this quality and Lan, Mat and Perrin have all unwittingly received followers for it. Granted, Mat IS that lucky and Perrin can talk with wolves, but hey...
Near the end of The Princess Bride, Inigo Montoya and Fezzik face a seemingly impossible task and feel lost without Vizzini, who was the brain of the outfit. They decide that the one they need to help them is the Man in Black—because he already climbed the Cliffs of Insanity without a rope, and beat the world's greatest fencer Inigo in a swordfight, the world's strongest giant Fezzik in wrestling and the, uh, kind of smart Vizzini in a game of wits.
The Brave Little Tailor, in The Brothers Grimm story, is a Type 4. Having killed seven flies, he embroiders a belt with the Badass Boast "Seven In One Blow" and then has to use his wits to fulfill challenges from people who think it was seven men.
Gaunt's Ghosts would fall under the first category.

Live-Action TV

The Doctor. He flies in a time machine that can go anywhere and anywhen in the universe, has saved the Earth more times than he can count, saved the universe and all of reality itself repeatedly. He defeats intergalactic races of pure evil on a daily basis, thinks crippling dictatorships is a rather average outing, and can do all of this with a kettle, a piece of string, and a screwdriver.

"It's impossible!"
"How impossible?"
"Two minutes."

Stargate Atlantis: "I'm Rodney McKay. Difficult takes a few seconds; impossible, a few minutes!"
Stargate SG-1: SG1 have gotten so adept at this the Asgard have, on more than one occasion, 'hired' them.
- There is even a sprinkle of this within (but not yet recognised outside of) the Destiny 'Expedition Crew' in Stargate Universe, especially for Eli.
In an old Polish series Zmiennicy, one of the workers at a theater says: "I do the impossible offhand, for miracles you have to wait a bit."
The Impossible Mission Force in Mission: Impossible.
Firefly: Used by Mal Reynolds during the Battle of Serenity Valley to describe his unit:

"We have done the impossible. And that makes us mighty."

Jack Bauer of 24. Throughout the series he persevered through gunshot wounds, beatings, torture, explosions, plane crashes, drug addiction, biological weaponry and even clinical death.

Video Games

The Master Chief from Halo is a Training from Hell example who later becomes an example of number two. He starts off as just being the only supersoldier available but as the series goes on it becomes clear that even for a SPARTAN he's an exceptional fighter.
- Similarly, in the earlier Bungie game Marathon, the character is implied to be a cyborg super-soldier, the hero of a thousand faces, and the personification of Destiny itself. As the series progresses, he goes from surgical strikes to being the turning point of a war to single-handedly routing entire armies.
The Prince in Vampire: The Masquerade Bloodlines wants to kill off the PC, but since he can't order an execution sends us on increasingly dangerous suicide missions.
Deconstructed in BioShock, where the protagonist was genetically designed to cut through Rapture like a hot knife through butter. The Vita Chambers help.
The arcade mode in Ace Combat 5 The Unsung War applies this to Mobius 1, the player character of the fourth game. The mission commander explains that Mobius 1 was chosen to put down an Erusian rebellion by himself because his kill count exceeds that of entire squadrons. Oh, and the Erusians know him as "the Grim Reaper."
Played with in Mass Effect. Shepard, on his/her own, is a clear type two, and was selected as the first human Spectre specifically on account of being a Badass - what he/she did, the player chooses at the character creation screen. Shepard's squad, on the other hand, tends to be closer to type one; most of them aren't as obviously Badass as Shepard when they first join, but by the time the games are over, they've done enough Level Grinding that impossible tasks are just another day's work.
- Played with by giving you a whole squad of Type 2's in Mass Effect 2. Every squadmate is either the best in their profession or just ridiculously tough - Zaeed in particular is described as just like Shepard except for the fact he takes cheques. According to the squad, they're also sent into suicidal odds about twice a day.
Airforce Delta Strike: Need a team to fly into the eye of a tornado to destroy its generator, destroy rail-gun artillery by flying down the barrel, or clear an absurdly spacious subway of enemy weapons? Call in the Delta Squadron.
Lightning almost invokes this trope word for word in Final Fantasy XIII, declaring she, the other party members, and humanity as a whole to be Type 2. By this point the gang has taken down multiple god-like beings, some of them had to be defeated several times, and the party is still going.

"We live to make the impossible possible!"

Type 2 is almost exactly how Samus is characterized in the manual for the original Metroid, having completed countless missions that others would have deemed impossible.
- Currently^[when?], she's pretty much a Type 5, having blown up Zebes, SR388, The Space Station circling SR388, The Bottleship, and the Phazon Planet.
Ace Attorney: In a less physical version of this trope, Phoenix Wright makes a living of turning an impossible trial on its head. If the only choices he has are the impossible and the even more impossibler, he'll do both simultaneously.

Web Comics

The gang of Exterminatus Now are an example of Surprisingly Elite Cannon Fodder. Schaefer has tried and failed to get them killed so many times on minor missions that he goes to them now if he really does have a job that needs doing against all odds.

Web Original

The Fellowship of The Questport Chronicles. Especially after Year Four, when they accomplish a task that is explicitly stated to be impossible.

Western Animation

Mickey Mouse in "Brave Little Tailor" whose boast that he "killed seven [flies] in one blow!" was taken out of context and he was tasked with taking out a nearby problem giant.
Kim Possible. "I can do anything" is her tagline. She's even got a song about it.
- Heck, her name alludes to this ability, which comes up as a semi-regular Phrase Catcher; Sometimes she'll do something remarkable or say she's planning to do so, and another character will say "It's impossible!". Ron's responds "Check the name." (About the only time the punny names are noticed in-universe.)
Phineas and Ferb, to the point that they just sort of assume it.