Brave New World: Leo himself said making a "Enhance Button" is impossible but "I figured out how to make one of my own in spite of that because I'm awesome." This disregard for physical laws makes Pikachu's eye twitch.
Nobody Dies: Jet Alone Prime generates an AT-field despite not being an Evangelion and Mana uses it to collapse an Angel's AT-field.
RE-TAKE: Shinji swats Arael out of the sky with his AT Field. Yes, a purely defensive ability somehow turned into a weapon.
My Little Denarians: In The Dresden Files canon it is thought impossible to destroy a Denarian's coin. The most you can normally do is kill the host and seal away the coin. Harry is understandably shocked when Lash's defeat of Lasciel results in the coin being destroyed.
Dexter: Also, battles will almost invariably end in some improbable manifestation of power.
In Elementals Of Harmony, Spike's fire breath apparently burns faster than entropy. The story acknowledges that this shouldn't make sense, but dragonfire, like dragons, is magical enough to make it work regardless.
Game Over: SC does do some things that circumvent other's expectations as to what is possible, although there's always the question of whether or not it always was possible and the other people just didn't think it was. SC's claim is that "anything is possible... except beating Kat."
"Gensoukyou Great Sports Day - 200 Yojana in a flash". At the start, Youmu's power just lets her run very fast but she lattter uses the same spellcard to exist outside space and time; the author even put in one of those stock A God Am I scenes.
Planeocracy: The Big Zam's I-Field neutralizes the attacks of Lyle and Katina... until Cherudim went Trans-AM and Lyle fired a concentrated sniper rifle/rifle bit shot at the field's generator. Now, keep in mind the I-Field neutralizes beams. It punched through.
Waking Nightmares: Harbinger qualifies becuase he's an earth pony and earth ponies do not have special powers like the unicorns and pegasi. Yet, he's faster than a train.
Time Lords and Terror: There was a storm in the first chapter that was "outright contradictory to all known magical laws." Trying to quantify the impossible event that happened over her head makes Twilight shout “AAAAAARRRGH!!!!”
In Stardust, Twilight's abilities break all the laws of physics Vahlen knows and their usage doesn't register on any of XCOM's sensors.
A My Little Pony: Friendship Is Magic fanfic titled 'Titanomachy' features Twilight hatching a plan that only the most desperate pony would try, using a formula that involved quantities of magic involving a constant she had computed but was certain wasn't done right, as it was 'in the range of the speed of light, in pony-lengths per second, times itself - a literal unbelievably large number.'. Luna later comments on this. Two stories later, so do the Fates, and they're not very happy about it. It involves basically jumpstarting her own corpse with the leftover corrupting essence of an evil god. So yes, she is in a way rather desperate at this point.
This is hilariously common in The Infinite Loops, due to Loopers applying solutions from one universe to problems in an entirely different universe which bypass the given ruleset making something impossible. Special mention, however, goes to a non awake Dr. Wily bypassing the antitech barrier in The Conversion Bureau setting with his army of robots; not destroying it, just bypassing it.
In many Naruto fics, like this one. Despite being sealed, Kurama can alter DNA and create new bloodlines.
Facing The Future Series: Danny's feral Super Mode gives him enough power to break free of Vlad's spectral energy neutralizer box that he normally couldn't escape on his own.
Thousand Shinji: According to the Magi, the data Ritsuko enters to determine how a massacre of black ops troops occurred is completely impossible. She has to write a patch for the software just to make the computers accept what they view as nonsensical data. (In reality, it just requires technology from 38,000 years in the future).
In Hybrid Theory, beyond the natural reality of Chi and the common impossibilities of Magic, you have the Third Circle, where people in moments of desperation can push themselves beyond all normal limits. Its presence can set people outside the rules and allow them to force new Concepts to be and always have been at the expense of collapsing timelines and accruing Paradox.
Thanks to Ukyou's constant exposure to the Third Circle due to her bond with Aaron, when Sailor Pluto freezes time in order to kill her, the ninja isn't affected - not through an act of will, but simply because Ukyou wasn't paying attention.
In Monsters In Paradise, this is Tobias's rationale for stocking his team with Legendaries at the Sinnoh tournament, as he feels that it's the person who can achieve the impossible that deserves to be the Champion. Whether he was referring to himself or to Ash's efforts against Darkrai and Latios is left unsaid.
The Boy Who Fell from the Sky & The Girl Who Found Him: When Naruto tries to restore Bolin's Earthbending with his Six Paths PowersNaruto some how turns Bolin from an Earthbender to an Airbender.
Rosario Vampire: Brightest Darkness: In Acts V and VI, Babylon's powers enable the souls of the dead to leave the afterlife, which the angels of Heaven insist shouldn't be possible.
Gabriel: Whoever this monster is, he... defied the laws of life itself.
In the Tengen Toppa Gurren Lagann-inspired Chrono Cross fanfic Heaven-Smiting Masked Magician, it has been stated repeatedly that it's impossible to destroy the incoming meteor in space for three reasons. First, whatever that tries to get past the atmosphere will get burnt up unless the object is extremely fast. Second, there's no oxygen in space, while there's no sign of spacesuit in-universe. Third, the meteor is over a mile in diameter. Guile, whose flight speed is just slightly faster than a hunting bird and armed with only his magic power and Boom Stick, doesn't care and brushes it off with a Badass Boast (complete with Kamina's pose) when the fortuneteller warns him about it. Through blasting the ground with Kamehame Hadoken, he gets past the atmosphere, and he later does the same trick on the Moon to get a speed boost in space, so that he can complete his mission before the oxygen supply in his body depletes. He succeeds, and his achievement terrifies people.