Near the climax, Beast, in order to find Belle, tears through dimensions by sheer force of will. Maleficent comments on how this was impossible; everyone else (herself included) either needs aid from the Heartless or a gummi ship to cross dimensions. Beast doesn't know how he did it. He just wanted to see Belle that badly.
Iji: Assassin Asha can dodge almost all attacks you throw at him on his second fight, including a nuke, even though you're fighting him within an enclosed space. A nuke. A nuke that, in his previous fight with you, triggered an instant victory.
Ar tonelico has three examples and justified in all of them:
Melody of Elemia: It is technically possible for a Reyvateil to craft a song outside her cosmosphere and without the aid of a Hymn Crystal, but it's so difficult and happens so rarely that anyone who sees it happen will say it's impossible.
Ar Tonelico 2 Melody of Metafalica. The game likes to mention how only one person at a time can dive into a single reyvateil with a two person dive being a big deal that only the most skilled therapist can perform. Jacqil says Frelia can handle four people at once. Everyone shouts how impossible this is, and normally they would be right, but Jacqil reminds them that Frelia is a goddess and so can handle it easily. Croix' response is essentially, "oh yeah....I forgot..."
Ar Tonelico 3 Knell of Ar Ciel: When the Rinkernator is open, Reyvateils can't use Song Magic because the source of it is disabled. Coccona can cast anyways because she is registered to an entirely different server called Infel Phira instead of the Third Tower whose Rinkernator was open at the time. There's a cutscene showing the energy transfer through space.
One fact that you're unlikely to discover during the main quest: you can break the cap on HP and MP! Granted, it's only by one digit each.
The first time you discover that there is such a thing as an Unsent, and that people don't necessarily turn into monsters after they die and go unexorcised, it probably seems like this trope. But then, the story goes on to break that rule so many times and in so many different ways that things veer toward a different trope entirely.
Battle Clash: Eddie and the ST Valius. This guy is so fast he can jump out of the corner of one side of the screen and reappear from the other less than a second later, landing in the same exact spot. In other words, he manages to cover the entire span of the stage you're on in a single jump. The stage in question (the Tower of Babel) is a giant space elevator.
In Borderlands, Some class abilities allow more ammo to be used than some guns can hold. In some cases this allows for five shotgun rounds to be used when onlytwo are loaded as well as sub-machinegun and assault rifles being able to hold as much as THOUSANDS of rounds.
In the sequel, Gaige has an ability called Anarchy, which increases her damage but reduces her accuracy for every stack. Under normal circumstances, the max number of stacks is 400, which equals 700% bonus to damage and an equal penalty to accuracy. However, with a special class mod that drops from a raid boss, you can get up to 600 stacks, for 1,050%. Once you pass about 550 stacks, bullets stop making any sort of sense. They'll zig-zag in mid-air, land behind you, bounce up and down, and all sorts of crazy stuff. In the rare event you actually hit something, they'll take so much damage, Ludicrous Gibs is almost guaranteed.
In the same sequel, Salvador's "5 Shots or 6" gives him a chance to gain 2 ammo upon shooting (1 is automatically consumed via said shooting, the other is added to his magazine), maxed out at 25%. Class mods (Most notably the Legendary Hoarder) can raise it to 50% or even 55%, meaning he's capable of spontaneously generating more ammo just by shooting. Vladof Rocket launchers don't consume ammo every 3rd shot and the Butcher shotgun randomly generates more ammo every few shots or so, meaning with these guns, Salvador can literally have his ammo gauge bounce from nearly empty to max, empty again, and keep going up and down. Once you reach this level of insanity, the best way to conserve ammo is to shoot as fast as possible and as much as possible.
In Asura's Wrath, at one point Asura becomes so mad, he gains the level of power it took his fellow Deities nearly 12000 YEARS to accumulate. Yasha even states that this is impossible by the standards they were using.
Similarly, killing Chakravartin was said by Chakravartin himself to be an impossibility.
Deus even uses this trope as his motto:
Deus:"Power without a purpose is meaningless and worthless. A purpose that is firm can change the impossible to the possible."
Almost all of Cerberus of Mass Effect 3 is semi-huskified and heavily indoctrinated. Check out their reaction to Shepard showing up on Mars;
Now realize that Shepard is actually scaring the hell out of people who can no longer think or feel anything for themselves. They merely duplicate the thoughts and feelings of those who control them. So Shepard "merely" strikes fear in the hearts of the Reapers.
At one point in Mass Effect 3, Garrus and Legion are debating weapon calibration. Legion insists that the maximum possible improvement is .32% percent. Garrus tinkers with the weapon, and manages to pull .43% improvement, baffling Legion. To drive the point further, Legion is an extremely sophisticated AI that is able to link directly to the machine. Garrus is 'just' a skilled technician.
I'll be...You've just redefined what's possible!!!
There's an implicit example in the epilogue, where MegaMan and ProtoMan's fight is dubbed "Tera-class". Those familiar with the series should recognize the fact that the series Battle Chip Tiers don't go beyond Giga.
Mother 3: Meta example. The online readme, about halfway down the page has a list of every hack that had to made in order to make the translation work. Buried in there is this gem:
The game only had enough memory for 40 letters of battle text on the screen at once. Fixing this was assumed impossible, but a fix was figured out anyway, allowing for infinite text per line.
Pokémon Colosseum: Duking finds Wild Pokemon in Orre, the desert land where no Pokemon can exist except imported ones. Normally this would not be impossible (strictly speaking) but the fact that no wild Pokemon exist in Orre is the defining aspect of both the story and gameplay.
The Legend of Spyro: The Eternal Night Post stage 1 of the boss fight. Scratch and Sniff manage to somehow pick Skabb off the ground and carry him out the window despite him being gigantic compared to them. Spyro and Sparx lampshade this.
Breath of Fire: Dragon Quarter: In order to finish off Chetyre, Ryu uses D-Breath to push his D-Ratio up to 169% and counting, when normally hitting 100% causes a Non-Standard Game Over due to his body being taken over by Odjn: the reason why he still manages to survive afterwards is because Odjn severs their bond afterwards.
Double Subverted in Sonic Colors, Sonic (who can run at least faster than the speed of sound), is able to outrun ablack hole (remember, this is something that's gravity prevents anything -which, if close enough, even includes light- from escaping from it). He resisted it for a good 30 seconds, and was almost sucked back in. Even Tails is amazed that he made it back to their world in one piece.
A character in Fallout: New Vegas claims to have done the impossible and shot a Brotherhood of Steel Paladin through the eye hole of their Power Armor. If you've met the Brotherhood and gotten in their good books you can challenge these Blatant Lies by pointing out that their eye protection is bullet proof.
Halo: If the ONI officer reviewing Cole's life is correct, Cole SURVIVED by executing an in-atmosphere slip-space jump, and is most likely finally living a life of peace far outside UNSC or Covenant space. He deserves it.
Myst: Yessha's mother Catherine was good at this. In the Novels she's mentioned as having written "Torus", an stable doughnut-shaped Age, which features a huge waterfall that falls through the planet's core, turns into rain and gets carried back by clouds to refill the Ocean, that in turn, feeds the Waterfall. Atrus' reaction to first seeing this was that until then, he had thought it impossible to do such things with the Art.
Tomb Raider: Jacqueline Natla certainly gets her money's worth out of her immortality which explains how she survived things like a pillar crashing on her or plunging into lava, but at one point she had an artifact that was capable of killing gods thrown at her. It didn't work.
Digimon World Data Squad: Yuma managed to make the physical embodiment of Lust feel pure love.
Dwarf Fortress The 'Death Gate' succession fort has a Battery that produces power from nothing. "The physics is weeping".
Several long-running succession forts (including Failcannon) have well-documented cases where the fortress's layout is so labyrinthine, so complex and bizarre, that it actually breaks the game's pathing engine. (In the case of Failcannon, this saved them from the wrath of at least one Forgotten Beast which wanted to ascend to the populated levels, and knew a path existed, but couldn't find it.) As if that weren't impossible enough Bizarchitecture, rooms sometimes exist that none of the players can locate. They can scroll without impediment throughout the entire fortress, and yet the only way to find these rooms is consistently to wait for a dwarf to come in and then zoom to them. Somehow, looking for it any other way, it cannot be found...
Super Robot Wars Alpha: In Original Generation 2, during the Dygenguard's debut, the machine fails to use the default TC-OS operating system. Filio reboots and reprograms the actual JINKI-1 OS...in a matter of seconds, just before Vigagi can get a bead on Sanger. Jonathan Kazahara lampshades on how it shouldn't be possible to do what Filio just did.
Pirate101: El Toro uses a whip for attacking and swinging over obstacles. He can even latch onto thin air with it.
In Shogun: Total War, In the Mongol Invasion campaign, you cannot perform a ceasefire from both sides- The Mongols neither have a Leader active and cannot make Emissaries. You can, however can stop the "At war" status on accident, by utilizing a Rebel revolt and not having any other borders touching the Opposing side. An example from the mongol side: ◊
Street Fighter: Gouki/Akuma hits Gen with the Shun Goku Satsu, an attack which attacks the very soul of the opponent, invariably killing them. Gen, despite having never experienced it AND being an old, sick man, managed to empty his soul in time to withstand it.
The main villain Sly Cooper and the Thievius Raccoonus, Clockwerk, has been in a feud with Sly's family for millennia. Turns out he turned himself into a robot that runs on the power of his own hatred in order to live so long. This is closer to Magitek than the rest of the more mundane science in the rest of the series.
The Nameless One of Planescape: Torment, if eloquent enough, can argue a person into nonexistence by logically proving to him he cannot exist.