The medium of Western Animation is such that literally anything is possible. Beyond The Impossible, in this case, refers to those rare actions and events that do not conform to the previously established rules of its universe. Generally, the more realistic a work is, the greater the likelihood that something going Beyond The Impossible can be found. Cartoons like Looney Tunes would be exempt from this trope, because the characters and settings follow no rules except the Rule of Funny (a rule which cannot be broken, only ignored).
In Avatar: The Last Airbender, Earth Benders repeatedly demonstrate that they cannot bend metal. Magic A Is Magic A is so strong with this one that an entire episode was based on earth benders as POWs in a metal prison. This trope comes into play near the season two climax when Toph invents metalbending. Not only did this come in handy during the rest of her adventures, but it also made her Legendary in the Sequel, The Legend of Korra. She also passed the art onto her two daughters, Lin and Suyin.
The Legend of Korra: "It's not a full moon...how are you doing this!?" Korra when Tarrlok bloodbends her into submission. A panning shot later confirmed that the moon was indeed only a crescent.
Amon can resist bloodbending.Tarrlok's bloodbending. It's been shown already that the only ones who are able to resist bloodbending are fully fledged Avatars and waterbenders that are at least as skilled as the bloodbender in question. And even without considering that, it was also shown that Tarrlok was able to take down many of the strongest characters in the show, including Korra, using his bloodbending. Revealed to be a subversion, however, when Amon turns out to be Tarrlok's older brother, and consequently a more skilled waterbender, and bloodbender, himself.
Yakone, father of Amon/Noatok and Tarrlok, also demonstrated the ability to bloodbend on whichever day he pleased. In fact, he deliberately avoided bloodbending during the full moon, so that when he was brought to trial, he claimed he had to be innocent because, as everyone knows, the crime he was accused of was physically impossible. Sokka shot this one down by pointing out that there were many recorded cases of people manifesting odd powers unknown to traditional bending, such as Combustion Man.
There's her notorious disregard of the fourth wall. Outside of a one-off gag or two, Spike is the only other character to engage in this, and does so far less overtly or consistently.
In the episode "A Friend In Deed," Pinkie Pie has Fantasy Sequence done in felt as she runs through her checklist of friend-making activities. After coming out of the spot, she resolves to complete her list, and punctuates it by holding up a green felt checkmark from her Fantasy Sequence.
In the same episode, she also sprouts an extra pair of limbs, which disappear just as suddenly. ...Yeah.
In "Too Many Pinkie Pies," Rainbow Dash is trying to nap by the lakeside. Pinkie, who still wants to jump and splash around, compromises by doing a huge cannon-ball - only to slow herself down in midair just as she reaches the water, and gently float the rest of the way down. Rainbow stares and asks, "How did you do that?" Though it gained less attention, she did the same thing while bouncing on a trampoline at the beginning of "The Best Night Ever."
In "Magic Duel" she played multiple instruments at the same time without using her mouth. note It was magically removed earlier that episode and she could neither speak nor eat.
This carries into Equestria Girls, where it stands out even more because the human world is explicitly non-magical. Among other feats, she pulls one of the balloon patterns off her skirt and inflates it, teleports behind Twilight Sparkle so she can grab her from behind, and later drops down from the top of the frame, upside down. During the attempt to keep Twilight's crown away from Sunset Shimmer near the climax, she also grabs the crown from Snips by materializing behind him. As usual, no one seems to notice any of this.
Everypony thought that the Sonic Rainboom was impossible until Rainbow Dash performed it.
"Most people thought the Sonic Rainboom was just an old mare's tale but on that day, the day I discovered racing, I proved that the legends were true; I made the impossible happen!"
Sweetie Belle's status as Lethal Chef is nothing important but there remains her culinary impossibility:
Rarity: I didn't know you could burn juice.
Billy from Adventure Time. While the show has a habit of doing this anyway, Billy's backstory had him defeat the Ultimate Evil and literally PUNCH AN OCEAN TO DEATH. note Also, HE FOUGHT A BEAR!
Finn popped a bubble he had made into a black hole with a four-dimensional sword despite in show and real life logic and physics saying otherwise.
In Wakfu, Nox, the main villain of the first season, is a powerful Time Mage who has been alive for over 200 years, collecting life energy and feeding it to a powerful magical artifact called the Eliacube, trying to accumulate enough energy to turn back time in order to undo a mistake which caused the death of his family. He has slaughtered countless people and entire civilizations for this purpose alone, despite being warned that doing so is impossible, even for a Time Mage, and could instead result in reality being destroyed. In the end, he manages to achieve his energy goal and actually succeeds in turning back time by 20 minutes.
The Transformers: Alternity Megatron's Tesseractal Swords are blades of infinite length. "Physical space is not an obstruction."
Jonesy from 6teen can get a tan from the mall's florescent lights. His explaination: "The key to a good va-cay is all in the mind ." Nikki shouted how impossible this was but the tan was there so there wasn't much she could do about it.
Teen Titans: Cyborg once got in a Brawler Lock with an insanely strong robot who was working at full power. Cyborg won by going farther than full power. His meter was at around 160% before it exploded. This was in fact the aesop of the episode: Cyborg believed it was impossible for him to get any stronger/faster/better etc. because "I'm a robot; my limits are built in." His human half allowed his mechanical half to perform better than what should have been possible. note This has been discussed extensively. The short version: The "100%" was a measure of safe capacity, not maximum. He forced himself to exceed his safe limits.
Many of Professor Farnsworth's inventions involve screwing logic. For example, the Planet Express ship doesn't fly, it stays put and moves the universe around it. Not to mention the boxes in Universe 1 and Universe A, which contain each other's universes - until the Professor and his Universe A counterpart somehow switch them, so that each box contains its own universe. note as we understand the Theory of Relativity right now, saying that the ship moves the universe around it is a perfectly reasonable point of view. That is, in fact, one of the cornerstones of Relativity. But bringing Real Life physics to show that the joke is possible ruins the joke.
Professor: Nothing is impossible. Not if you can imagine it.
On multiple occasions, bending robots have bent "unbendable" girders.
This is why they've provided the page quote for Voodoo Shark.
Bender's arms came off in the pilot and he managed to put them both back on by himself. Lampshaded by Fry: "I don't know how you did that".note This is subtly explained in a later episode: Bender's arms are shown to be able to operate completely independently of his body, and seem to have some measure of artificial intelligence
Cosgrove, friend of Freakazoid!, has the ability to make any group of characters immediately stop fighting and behave themselves, from common burglars to supervillain mooks. It even works on The Warner Brothers. The premise of their own show was their inability to be contained or controlled.
There are techniques called the 'Six Impossible Moves' which would be better named 'six moves that are impossible for anyone lacking 'The Gift'. Po and Fenghuang can do them because they have it and this is presumably what fuels the latter's "strong rule the weak" beliefs.
"The Three Needles" is a trick that is repeatedly stated to be impossible. Po accomplishes it by cheating with magic shoes. Which is how Shifu and the others knew that he was cheating.
Bens City: "The Debris" contains a lot of supernatural elements, and the ending of "The All Inclusive" feels like a Karmic Twist Ending. Yet, most of the rest of the series happens in a very realistic world.
The first plot arc of DuckTales featured a dog(?) who had been looking for the city of gold for centuries. When asked how he stayed alive that long, all he said was "Sheer willpower!"