Impossible Genius

You know how you would hear jokes about how The Professor in Gilligan's Island could build anything out of coconut shells and sand? This is when you have a character (or group of characters) who can actually do stuff like that. It's Played for Laughs, but in-universe they aren't joking; the character or group really is that good.

Sometimes a characteristic of an Omnidisciplinary Scientist, Evil Genius, or The Smart Guy. Might involve Bamboo Technology. See also The Spark of Genius, Mad Scientist. Overlaps with It Runs on Nonsensoleum if the parts or the principles involved are sufficiently ridiculous. If it's made clear that the character's invention only works because they're making it work with magic or superpowers, that's Magic-Powered Pseudoscience.

Niece-trope of The Ace. While The Ace can accomplish any feat imaginable because he's Just That Awesome, these characters can do physically impossible feats of engineering and construction because they're Just That Smart. They may casually dismiss such feats as being (from their perspective) pathetically easy. See also Testosterone Poisoning, for characters who are Just That Manly. Compare Techno Wizard.

Anime and Manga

 * Pretty much a staple of all Shonen detective mangas.

Comic Books

 * In All Fall Down, both IQ and IQ Squared are these. The former built doomsday weapons, the latter created a software empire worth billions.

Literature

 * The Smarty-Pants from Gordon Korman's Nose Pickers From Outer Space books. The final exam for admission to their ranks includes memorizing the infinity times-table, and they've harnessed the power of the coleslaw molecule.

Live Action TV

 * MacGyver, although his reputation "aren't you that guy that can build a time machine out of three paperclips?" was far less realistic than the actual character.
 * MythBusters has tested a number of myths of this sort. In a number of cases they actually succeeded in accomplishing the seemingly unlikely in Real Life, for example, building a rope out of toilet paper, but usually only after days of work using large amounts of raw material.

Tabletop Games

 * The Izzet League from the Ravnica setting of Magic the Gathering are portrayed like this. Niv-Mizzet himself takes it up another order of magnitude.
 * Genius: The Transgression has players take the role of Mad Scientists who are capable of this at the cost of risking Science-Related Memetic Disorder.
 * Anyone with Quick Gadgeteer advantage in GURPS.

Web Original

 * The Webcomic Girl Genius has Sparks, hyper-geniuses that regularly laugh in the faces of all laws, human or natural, with their wacky creations.
 * Casey and Andy can build anything that does anything, as long as you describe it as a "[main purpose or function]-o-mat". At one point, Quantum Cop asked for their aid and described what he wanted the device to do; they refused since the idea was impossible. They had no problem at all when he corrected himself and just asked for the appropriate -o-mat.

Western Animation

 * In Phineas and Ferb, the title characters. Dr. Doofenshmirtz and Agent P also have a touch of this.
 * Egghead Jr., the small chicken in Foghorn Leghorn cartoons.