Death is more of a nuisance for Kamina who can infiltrate a Lotus-Eater Machine and escape with his buddies, despite the Anti-Spiral saying both of these events were impossible for 'intelligent life forms'.
Simon creates a mecha with Lagann that can punch an enemy to the end of the Space Time continuum. It makes this big hole in reality.
Attenborough can use laser cannons to shoot points in space time. Not just one point but every point. It's like he's pointing a gun at history. Then he goes and shoots every single point in space and time at once. They never made this weapon; Attenborough created it on the spot simply because he wanted More Dakka.
Viral is a beast man who was created to be absolutely physically incapable of generating spiral energy, but he becomes so Hot-Blooded he can do it anyway.
One of their battles was calculated to have a zero percent chance of success. They succeeded anyway, because they altered probability itself by sheer force of will.
Among Black Jack's many medical accomplishments are full-body skin grafts, arm transplants, brain transplants, fingerprint transplants, grafting two people together to share one heart, extracting full-body parasites, operating at lightning speed, operating blind, operating in space, operating on a dozen patients at once, operating on dogs, cats, deer, bears, monkeys, birds, whales, aliens, ghosts, mummies, plants and computers, and removing a parasitic worm from his own intestines while under assault by a pack of wild dingos.
Black Jack 21 takes all of this and turns it Up to Eleven. Example scenario: Dr. Black Jack must save the life of the pilot (critically injured by a point blank grenade hit in the cockpit) of a turbulence-ridden and heavily damaged jet so that he may guide someone else into landing it before the plane runs out of fuel and dooms all of its passengers. However, before BJ can do this, he must first get into the cockpit via an emergency exit that can only be accessed by exiting the passenger cabin and climbing atop of the plane's fuselage, all while the plane is still thousands of feet in the air and flying at full speed.
Baccano!: The rules of Immortality are spelled out and fixed but Fermet has defied one of the biggies. Namely that being eaten by another immortal is the only way they can die. He's recovered from it twice.
Also, Claire Stanfield. He thinks he's God, and considering the stuff he does he may not be too far off the mark.
Subverted with Aizen and the Hougyoku. At first, it looks like the reason Aizen wants the Hōgyoku because there are rules separating Shinigami from Hollows and he wants to break them. One example he uses is his claim that he can transcend Reason itself and demonstrates this by allowing Koutotsu (a being of Reason) to destroy itself just by coming into contact with him. However, this trope is subverted because Aizen himself admits the Hougyoku is incapable of achieving the impossible. It can only achieve that which is possible (no matter how improbable) to achieve without the Hougyoku's involvement.
Dragon Ball Z: The Supreme Kai thinks this when he finds out how powerful the earth Saiyans are. To explain: this guy is the series' equivalent of God and yet his power level is nothing compared to them. The saiyans outgrew him by refusing to accept a limit to their potential power. Goku, in his base and weakest form, is miles ahead of the Supreme Kai in terms of raw power. note Whilst he doesn't know how to use it to do anything other than blowing up, it's still power, and it's also the only reason why he's lower in the Super Weight class than the gods His son, Gohan, is even stronger after the oldest god in the universe unlocks his powers. Buu is also stronger than Goku. Buu proceeds to absorb Gohan, Goten, Trunks and Piccolo. With that much power, he can destroy the universe if angered into using it. So, you've got that written down? Now imagine that Vegetto, the fused form of Goku and Vegeta using the Gods' earrings, is even STRONGER than that... in his base form. Now try to picture how much real estate damage he could do as a Super Saiyan 3. Or worse. As a Super Saiyan 3 Giant Monkey. Gogeta Super Saiyan 4 is the fusion of Goku and Vegeta in their Super Saiyan 4 form, who are stronger than almost anything in the series. A flick of his finger could end the universe as we know it.
When Vegetto was turned into a jawbreaker by Super Buu, he revealed he could not only control himself in that state, but effectively fight him, and was even more dangerous as a candy drop, because of the small size and speed that made it harder for Buu to fight him. No other character has even demonstrated that they could become completely mobile when Buu turned them into candy. The implication with Vegetto is that he could by simply being that strong.
The main series has Kaito Kid's voice manipulation ability. He can perfectly imitate anyone's voice, even girls and young children. Conan requires a device created by a Mad Scientist to do that, and when he first witnesses Kaito's ability this is what he thinks.
Death Note: Light tries to do this with the Death Note's rules but Magic A Is Magic A and he can only fake it. It bites him in the ass later on.
In episode 5, we find out that The last space monster is trapped in a black hole, and is attempting to break out of it. Not only does it succeed despite Nono's best efforts, but it OVERCOMES the black hole in size and uses it as it's POWER SOURCE. Note that the black hole is the one created in Gunbuster with the self-destruction of the Excelion.
In the final battle in episode 6, Lal'c and Nono not only obliterate the space monster, which had taken a planet to the face and shook it off as if nothing happened, but they also CRACK a black hole. After that, what appears to be a giant pair or hands clasps over the black hole until it is subdued. Even the android military advisor on one of the remaining ships stated something to the effect of, "It is pointless to try and explain. They are now beyond our understanding..."
Summed up nicely with this line: The black hole is splitting....that's not allowed in this universe..."
Digimon Adventure commenced before the Mega level was introduced in the broader canon, so the level only first appeared within the last third of the series; as such the series generally called the Ultimate level the strongest level and its members "fully-evolved" and "strongest Digimon". It was introduced by Myotismon, already an immensely powerful Ultimate who had already singlehandedly overwhelmed the heroes several times before being killed, being resurrected as the Ultimate-level VenomMyotismon and proceeding to wipe the floor with the heroes. Matt compares it to changing the rules partway through the game.
Season 1 - Two of the protagonists get shot by holy arrows.
Season 2 - The Digital World's equivalent of a god lends them their power.
Season 3 - The Digimon merge with their partners to reach Mega, and require the power of the aforementioned god to do it outside of the Digital World.
Season 4 - The angels that rule over the Digital World give them the power. Twice. Takuya and Koji effectivly had two mega levels (although they were called Fusion and Zeta Hybrids) and both times needed divine intervention.
It's established early on and confirmed near the end that N.O. capability humans need to be hit by Haruka's bat to use their heads as portals. That never happens to Eri, yet she summoned a Medical Machinan machine after headbutting Naota.The director's commentary explains it was because of how Naota and Eri knocked noggins when Naota is hit by Haruka's moped earlier. There's some kind of transitive property to it but still weird.
How does Canti fly in the second episode? It's like he took fake decorative wings and made them magic when no one was looking. "It flies around like its never heard of gravity."
Jack Rakan who does this regularly. For example: a particular villain had warped reality so that he ceases to exist entirely, prompting a Heroic BSOD and Super-Powered Evil Side relapse in the Kid Hero. Rakan immediately wills himself back into existing temporarily, for the sole purpose of smacking Negi upside the head. Chisame says he has 'infinite cheats'.
Vampires are stated to be weak to garlic but Evangeline adds it to her ramen. They're also weak to sunlight, and crosses, and running water, and all the other "traditional weaknesses"... there's a reason why one of Eva's titles translates as "Daywalker" and why she's the series' patron saint for Training from Hell.
A prerequisite of a Pactio is having a soul. Chachamaru believed she couldn't get a Pactio because she is an artificial construct. Either she really did have a soul, or Negi's magic is great enough to screw over the rules of the Pactio system and create a soul.
The Class Rep almost broke the Pactio system when she pacted with Negi; her love for him was so great she inverted the spell and nearly became the dominant partner in the contract, despite not having any magical power. This one of the only examples of love being quantifiably taken Beyond The Impossible.
Killer Bee can transform into his biju without going nuts, which the previous two jinchuriki (one of them The Hero) demonstrated to be impossible. At least until chapter 571, when Naruto tames Kurama and can transform at will. However, due to the bond being incomplete, Naruto can only stay in full Biju Mode for 5-8 minutes at most.
Kushina survived having Kurama extracted from her, something that was assumed to be universally fatal,
the Third Raikage No Selling a Rasenshuriken, an attack that, up to this point, had pretty much instantly disintegrated everything else it hit. Even the Kurama took more damage from it.
Chapter 533: Chouji grows chakra wings without using the three soldier pills in succession, a feat that blows away his own father.
During the Fourth Ninja World War, Naruto's Kage Bunshins become this trope. These are supposed to disappear after taking one direct hit, and yet, Naruto sends them to several battlefields where they take down powerful enemies, who are supposedly immortal.
Sanji can ignite his legs with stomping and not get burned. UNDERWATER, even. He kicked Duval pretty.
A sword ate a fruit that turns it into an elephant. This one is explained as a breakthrough by the series' resident Omnidisciplinary Scientist, Vegapunk. Oda has hinted that when he finally makes his appearance, a lot of questions will be answered.
Blackbeard managed to steal Whitebeard's Devil Fruit powers; almost two hundred chapters prior, it was stated that anyone who tried to use the power of more than one Devil Fruit would explode, and it still hasn't been explained how he did it.
Durarara!!'s Shizuo. First of all he's super strong in a show that (some elements aside) is pretty realistic. More impossible, he's broken so many bones its amazing he can even walk. His buddy Shinra, a doctor, stopped trying to figure out how he does it. At one point, he punches a thug so hard he knocks him out of his clothing.
The Slayers uses theurgic magic (meaning it draws from faith in Gods or Demons). Martina uses a curse, on raw willpower drawing from the energy of a spirit that doesn't exist, since she made it up. Granted, it isn't nearly as impressive as nearly anyone's magic, but that she can use this at all, even for minor curses or dowsing is insane.
Homura is an irregularity among Magical Girls because Kyubey can't remember making a contract with her. Kyokou notes that this should be impossible. Indeed it is. Homura made a contract with Kyubey in a previous time line.
In Crest of the Stars Ekuryua's driving is so bad that she induces motion sickness in a shuttle full of Abh. This is supposed to be impossible because Abh are genetically engineered to fly in ships without inertia dampening.
Kenichi: The Mightiest Disciple: Sakaki Shio, the "100-dan" karate master of Ryozanpaku, versus Akira Hongo, the "God Fist" of YAMI's One Shadow Nine Fists. They engage each other in battle and manage to knock each other unconscious at the same time after a grueling fight. They keep right on fighting DESPITE BEING COMPLETELY UNCONSCIOUS.
Marcus from Digimon Savers once broke an unbreakable shield with his bare fist.
Luke in The Sacred Blacksmith specializes in forging katanas. Eventually he forges a katana that is so perfect, so sharp, that he sticks it into a table and orders someone to take their sword and strike the katana. Remember, this katana is at rest. Absolutely no motion or force is behind it. The other sword is swung at full force and its blade gets sliced cleanly in half without so much as leaving a scratch on the katana. Impossible would be the Katana surviving the strike. In this case, Beyond the impossible is the striking weapon being cleanly destroyed without so much as a resulting blemish in the effort. This isn't even one of his magically forged ones (those break pretty easily). This one was made the mundane way with mundane materials and yet it still does the above.
Despite a lot of extreme and reality bending abilities, Aka Aoki has the distinction of an Abnormality that is impossible according to the series' own logic. She can infect any person with any sickness ...and she uses it to cure any damage done during Flask 13 and the subsequent trials.
Iihiko kills Anshin'in even though some of her skills include invincibility and guaranteed victory against a stronger opponent.
Issei has a Power Fist that doubles his power every ten seconds, but resets automatically every five or six boosts according to how much power his body can handle. Then Vali unintentionally threatened to cut Rias's bust size in half prompting an Unstoppable Rage and dozens upon dozens of boosts far beyond Issei's previously-established limits.
There's also Issei accidentally summoning the breast god whose existence was completely unknown to other gods.
Fushigi Yuugi has a place in Kotou called the "Shrine of Seiryuu"; a Physical God that stands opposed to the god Suzaku. In this place, the warriors of Suzaku are forbidden entry by a magic barrier that specifically targets them and the priestess of Suzaku is paralyzed. Chichiri can get in nonetheless because "I'm trickery, you know?" while Tomahome is powered by love. To a native of the Universe of the Four Gods this is as impossible as someone ignoring the laws of Thermodynamics in real life.
Shakugan no Shana. Tenmoku Ikko can cut Fecor's Magnesia with the Nietono no Shana; a feat that no Crimson Lord has ever matched. He's just a torch with a treasure tool!
In World Record, Dan manages to run fast enough to cause glitches in the Matrix as it struggles to keep up with him. Even when the Matrix freezes the program, it doesn't stop him for long.
After the Machines manage to reconnect him and make sure he can never walk again, he briefly manages to glitch gravity and begin floating.
In Kid's Story, The Kid woke himself up from the Matrix, an act nobody considered possible.
The Air Tracks of Air Gear strech the possible to its breaking point but it's subverted. It is easy to forget that tricks only seemed to be able to grant users flight, in the beginning of the manga. Over two hundred chapters in, and everyone with something roll-capable (including soda cans!) on their feet seems to put airbendingto shame. The real kicker near the end of the series. Clashing the energies of the Sky and Storm Regalias somehow disrupts gravity for a moment on Earth that people and objects temporarily float in the air. They defied the laws of physics to do this.
Card Captor Sakura: The price for sealing card #53 'Void' was the 'precious feeling' of the most powerful mage. This ended up being Syaoran due to the other candidate being exhausted at the time. The card was sealed and transformed but Syaoran didn't forget after all.
Code Geass: Emperor Charles said it was impossible to geass the world's equivalent to God, and when you think how Geass is refered to as 'the power of the king' it makes sense. Lelouch thought otherwise.
Cyborg Grandpa G doesn't agree with your interpretation of the word "impossible".
MÄR Phantom is the Number 1 Chess Knight for a reason. It becomes apparent during the fight with Ginta. He is shown to be capable of summoning up to five Guardians at the same time, a feat unaccomplished by anyone else ever. Dorothy said outright that this is impossible, and Alan equated the feat to writing two dissimilar letters—one with each hand. Most people can only summon one Guardian due to the incredible stress on stamina, mental strength and magic power. Not to mention you can't move much while one's out. And he isn't restrained to using techniques of this level only once during the battle.
Rio -Rainbow Gate!- : The Sky Resort, a building that flies without any visible propulsion, defying all kinds of physics and gravity laws.
Seitokai Yakuindomo: The second half of episode 13 looks like it's going to be the student council reflecting on their memories of the past year, but it quickly turns absurd when they flash back to events that never happened, never can happen and, in the case of Mutsumi, are in a completely different genre. It gets weirder: apparently Tsuda's souvenir changed to conform to Shino's warped memory, causing Tsuda to go into a short Heroic BSOD. In other words, it literally WARPS REALITY, which flies in the face of the rules of the rest of the series which is otherwise mostly a Slice of Life.
Campione!: One of the main constants throughout the series is that normal humans cannot stand up to gods. Youth Form smashes this apart, as those given Protection have the same overwhelming life force and magical capacity as a Campione. Erica was able to defeat the possessed Ena using it.
Hikaru no Go: Downplayed for a mundane example; Akira playing four different games at once and ending all of them with tied scores. Deliberately achieving a draw with an opponent that is not also trying to do the same is one of the most difficult things to do in Go, and the skill level of your opponent makes no real difference. Akira doing this in four simultaneous games is treated as a superhuman achievement.
Witch Hunter: Like Tasha's Song of Moirae, Ryuhwan can access a powerful, albeit weaker, form by allowing a large amount of mana to enter his body. Unlike Tasha's though, he can enter this state without being driven insane but can only use it in a limited amount of time before his body rebounds from the strain.
Beelzebub has Toujou; a completely normal human, no special powers or contracts, capable of fighting toe-to-toe with the most powerful demons. At one point, a demon (who is used to fighting other demons) uses a hypersonic sword to pull a Single-Stroke Battle on him. Except Toujou blocks and breaks the blade with his bare hands. The watching demons are stunned nearly speechless, and decide that they (read: powerful soldier demons trained to fight other powerful soldier demons) need to team up to fight him. He defeats them offscreen.
One Punch Man: Saitama's strength training is absurdly simplistic, but it made him able to defeat skyscraper-sized monsters with less effort then Superman does. The other characters who have heard his explanation comment on how flat-out impossible his strength is when compared to what he claims to have done.
His speed is so far beyond super-human it is ridiculous. During the first fight with Speed of Sound Sonic, Sonic performs a series of high speed dodges that would be impossible for any normal human to see, let alone keep up with. When finally Saitama decides to move, he moves so fast that even Sonic can't see it.
In A Certain Magical Index, Fremea Seivelun manages through sheer willpower to hack the entire Academy City's AIM field to both end the Agitate Halation effect that was driving many of the heroes crazy and turn an AIM thought being vulnerable. These feats are supposed to be impossible because Fremea is a Level 0 esper; someone with no power.
Sword Art Online notes several instances within the otherwise solid consistency of this death-game RPG's rules: When you die in game, you die in real-life, period. However, both lead characters have occasionally bypassed the rules of the game through sheer will, performing game feats that no others can do, albeit extremely rarely. Coming back to the game after being killed, to note one instance.