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.
Super Robot Wars Unlimited Generation provides this: Shiniji Ikari broke Eva's laws of physicsto save Rei from death. All because he simply cared about her so much that he didn't give a damn what limits were in his way, including fate itself.
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. Kurama despite being sealed, can alter DNA and created 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.
According to the Magi in Thousand Shinji, 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).