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Black Mage's Evilness. When the representation of all of your sins is yourself, because nothing else was quite up to the task, and yet you manage to be even WORSE than it, then RAPE IT, and EAT IT...
Also, Thief steals his own class change. From the future. At the worst moment possible, to boot.
Fighter uses this trope to prevent the Light Warriors from dying after a long fall. How? He blocked the ground with his sword. Don't ask him how he did it; somehow it will make sense.
Black Belt's No Sense of Direction breaks the laws of time and/or physics on a regular basis. For instance, crossing a tightrope that's only taut because he's holding one end of it.
Problem Sleuth broke the impossible way back when they encountered a Solve the Soup Cans puzzle so utterly impenetrable and difficult that the protagonist was forced to reload from a save-state made after the puzzle was completed.
"Shit is more real than it has ever been or possibly ever can be."
At one point in the story, Jade promises Echidna to complete a task that she believes is impossible. That is, she must bring all the lands, denizens, consorts, and everything else with her when the kids escape the universe and the Scratch. At the end of act 5, after reaching the god-tiers, Jade shrinks the planets themselves into convenient travel-size, and makes it clear that the "impossible" is now trivial.
Sollux Captor once hacked Doc Scratch's typewriter.
The Autoresponder doesn't show off just by reciting pi to an excessive number of digits, but by claiming to have calculated the final digit of pi (four, apparently). This is obviously impossible (pi is an irrational number that has no end) and he's only saying it to screw with Dirk, as Dirk rightly points out.
During the end of Act 6 Part 3, Lord English kills the doomed versions of the Beta Kids and Post Scratch trolls and, for lack of a better word, breaks the afterlife. Absolutely no one is immune to being killed off by Hussie. Not evenHussiehimself.
Girls Next Door uses this trope to explain why Chihiro is the unofficial president of the Wibsy (Weirdo Boyfrend & Supernatural Stalker club): using Christine words "It's hard to trump dating a river." The others have at least human-shaped (a fae, a ghost and a vampire amongst others. Christine with her murderous deformed psycho is the most normal of them all.) boyfriends/stalkers, so this is impossible by their standards. in her defense the river was in the form of a human boy that looked her age at the time she fell for him.
Freefall: At one point Sam and Helix go to return an authorized (as in, legally produced) autographed (as in, signed by the creator) copy of the The Star Wars Holiday Special. George Lucas considers the film such an Old Shame that he's taken steps to ensure that it will never be legally produced while he's still alive to sign it.