Most of the main characters in Naruto, starting with the Uchihas (yes, all of them) and former Big Bad Orochimaru and working from there. It's really more notable when characters considered "geniuses" don't turn evil than when they do. The most notable of which being Itachi, who it turns out was on the good side all along even though he is arguably the most gifted.
Kongo Agon of Eyeshield 21 is another advanced example. He's described as a once-in-a-century football prodigy, capable of reacting in 0.1 second to any move the opposing team makes, as fast as physically possible. He never goes to practice and never exercises, at all, in sharp contrast to his brother Unsui, who trains tirelessly every day but can't keep up with him. Deimon managed to throw Agon a loop by sending in benchwarmer Manabu Yukimitsu, who managed to overtake Agon through sheer determination and tenacity. Oh, and he's practically the Anthropomorphic Personification of the Opposing Sports Team.
Brooklyn of Beyblade fits this to a tee. Thanks to a savant-level natural talent at playing spinning tops, he's never had to try in his entire life. On the one hand, far from making him an Arrogant Kung-Fu Guy, he's actually a pretty nice guy who approaches the game with a zen-like calmness. On the other, when he's finally defeated for the first time, he actually starts trying... and the results are not pretty at all.
Yujiro Hanma from Baki the Grappler is by far the most talented character in the series. He was born with so many gifts his birthday might as well be Christmas. He has become increasingly bored in his later years since no one can offer him a decent fight. He became so desperate that he fathered a child and put him through hell in the hopes he would offer a challenge. Yujiro even threatened his son Baki that if he doesn't live up to his expectations that he would repeat the whole process over with Baki's children
Negi Springfield is often called 'Prodigy' or 'a Genius' thanks to his ludicrous growth, amazing smarts and innate ability to master many fighting styles. Neither brash nor overconfident, he's instead insecure, self-sacrificing, somewhat power-hungry with a noted inferiority complex, which eventually causes him to choose the powers of darkness to prevent harm to his students, in that sparing himself unnecessary pain at losing more loved ones. His lonely disposition has yet to be fully confronted, though he occasionally gets accused of being a Martyr Without a Cause.
Later on we get introduced to Kurt Godel, who's also described as a prodigy/genius. Unlike Negi, he's a Smug Super and is the epitome of the Wild Card.
Berserker of Kenichi: The Mightiest Disciple poses the danger he does entirely because of this. Berserker is so naturally talented in martial arts that he can defeat the best of his peers despite not having had any formal training whatsoever. His defeat requires a massive aversion of Hard Work Hardly Works, with Hermit claiming to have put in 10,000 times the normal effort. Unfortunately, a later chapter of the manga suggests he has since received that formal training.
Puella Magi Madoka Magica puts a particularly dark spin on this. The titular character is noted early on to have vast potential as a Magical Girl. Too bad this means her eventual, inevitable transformation into her Superpowered Evil Side will destroy the world if she contracts and activates her powers. Despite the fact that she is a sweet, innocent, trusting person with a low opinion of herself, her Gift has the potential to wreak more destruction on the Earth than that of most other examples.
From Katanagatari, we have Yasuri Nanami, the main character's Ill Girl older sister. Let's count the ways: first, she's so naturally strong and talented that her father, the head of the most deadly swordsmanship school of the era, had literally no idea how he could possibly teach her. This, and not her frail body or gender, is the only reason she isn't the head of Kyotouryuu. Her body absolutely refuses to let her die, even though she herself would like nothing better. She can completely read the moveset and secrets of any martial art by seeing someone take a stance in it; and moreover instantly learns any technique by seeing it once, and masters it by seeing it twice (or performing it herself once) - even if that technique involves restructuring her own body on the fly. And finally, the ultimate goal of this utterly broken Mega Manning is to take her further away from using her own strength and skill so that Shichika can finally kill her, because otherwise he'd have no chance.
Bleach: Ichigo has an innate fighting talent that allows him to develop his combat abiilities at a truly phenomenal growth rate. He was able to become capable of fighting, and defeating, lieutenants and captains after just ten days training. He was able to achieve Bankai in two and a half days, and within the space of only a couple of months in total, he went from being a powerless person who could see ghosts to being capable of defeating a god-mode Aizen. All the various powers his mixed heritage has given him merely distract from the fact that his combat ability is completely innate. Various characters in the manga have lampshaded this, observing time and again that combat training doesn't work on him and that his ability to fight isn't evidence of heritage or training but of instinct. Even when he loses all his powers, his combat instinct makes him incredibly good at handling unsavoury humans such as thieves.
Akagi won his first match in Mah Jong at the age of 13 after a ten-minute introduction to the rules. A few hours later he had defeated a professional rep player for some Yakuza who was cheating and backed up by the other two players. Things only got better from there. The manga repeatedly references just how unnaturally skilled Akagi is at gambling in general, and particularly in Mah Jong, to the degree he's almost killed once for failing to lose. In Ten, where he has been a gambler for over thirty years, he is renowned for having never lost a game in his life.
Mikasa Ackerman from Attack on Titan is described as a prodigy of unparalleled talent, with a natural mastery in all her subjects. She's still outshined by in-series World's Strongest Man Levi, though, although it's possible she may grow past him with combat experience.
Hajime No Ippo There are three boxers in the series that are far beyond the rest of the cast:
Takamura is stated by the author to be the best boxer in the series and when he was found by his coach was considered a "diamond that didn't need to be polished". His fights are usually one-sided affairs and played for laughs but one of his opponents, Bryan Hawk, actually had more talent than Takamura; so much more that he gave Takamura his hardest match in the series. Takamura won, but it was partly because Bryan Hawk never trained for any of his fights and relied on his raw talent while Takamura alwaystrains his hardest for fights.
Woli from Indonesia. He is by far the character with the most raw talent, out stripping even Takamura and Bryan Hawk. It took him only three fights to become the Indonesian champ and he gave the main character, Ippo (who at the time had 22 fights and had defeated multiple national champions), a very tough fight. Ippo's coach stated that they would never fight Woli again as a second fight would end in Ippo's loss since the only thing Woli was lacking was experience.
Dragon Ball Vegeta likes to think he was born with the Gift being a prince from a race of proud warriors who boast about being the strongest in the universe. It does not help that he was more or less called special by all those who knew him, even his Bad Boss Frieza. He is not too far off. He is extremely talented and reach levels that none in his race could comprehend. However, Goku has a greater gift for fighting, being able to outpace Vegeta no matter how hard he trains. During the Cell and Buu Sages he states that Goku has a gift, something he denied before by calling Goku low class trash. By the end of the series Vegeta had to fully admit to himself that Goku was the superior fighter. Even then, both Vegeta and Goku's gifts are outpaced by Gohan who can easily become the strongest in the universe if he desired it.
Inverted in Marvel's G.I. Joe continuity. Snake-Eyes, a white friend of the family, is regarded by the masters of the Arashikage ninja clan as being their most gifted and promising student. He stays good (probably partly because the masters don't tell him this to his face), but his friend, Storm Shadow, a blood member of the clan who had trained with them since childhood, becomes embittered at being constantly thought of as coming up short compared to his friend.
Moondragon is a classic case of this. Orphaned by the renegade Thanos of Titan, she was brought up on his homeworld and instructed in physical and mental disciplines for which she proved to have considerable talent. She got proud enough to challenge the Dragon in the Moon and apparently proved good enough to destroy it, which did not help. At her best she is insufferably arrogant (being almost That Damned Good to boot), and when the DitM's influence surfaces she lapses into full-blown Megalomania. Life with her new girlfriend seems to have mellowed her... somewhat.
The Gorgon, of Marvel Comics, is a ridiculously advanced case. He could read and write by his first birthday. By four, he was one of Japan's most acclaimed artists. He composed his first opera at age six. At twelve, he wrote a mathematical equation that proved the existence of God. He became a ridiculously-skilled martial artist in adulthood, as well. He's also one of the most bugfuck insane guys out there, and is fanatically devoted to the evil Hand. Oh, and he can also turn you to stone by looking at you. His actions so frequently cause "WHAT NOBODY IS THAT [good/fast/strong/silent]!" reactions that "Wrong. The Gorgon is that [whichever previous adjective]" is practically his Catch Phrase.
Film - Animated
There are more than a few echoes of this in how Master Shifu treats Tai Lung in Kung Fu Panda—and true to form, seeing only the snow leopard's incredible natural talent for kung fu (he was after all the only one to master all one thousand scrolls) the guy proudly pumps him up to be the Dragon Warrior, all without seeing the darkness that was growing in his son's heart. Though Tai Lung turned evil, he luckily didn't indulge in a great deal of Wangst. You can guarantee, though, that if he didn't die in the final battle, shows up in the sequel, and does a Heel-Face Turn, he will become either an Ineffectual Loner (which he may well have been before his Start of Darkness) or thanks to Defeat Means Friendship, an Aloof Big Brother to Po.
Remy of Ratatouille is something of a subversion: he possesses the gift of incredible cooking skills, but unlike the examples here nobody looks up to him or is envious of it because he's a rat, who don't need to cook and aren't allowed in kitchens anyway. When he's able to express his gift by being The Man Behind the Man of a human, he gets taken advantage of: his rat clan uses him to steal food from the kitchen while his human "puppet" (who couldn't boil spaghetti without Remy's help) takes all the credit.
Played with in The Matrix. The Oracle's comment to Neo about him not being the One ("Sorry, kid. You've got the gift, but it feels like you're waiting on something.") may seem like a confirmation of Neo's fears, but it actually leads into some foreshadowing of his eventual fate ("Your next life, maybe. That's how these things usually work." and "One of you is going to die." and can be seen as clever manipulation to hook Neo into the path she wants him to take.
Film - Live-Action
Sing, the eventual protagonist of Kung Fu Hustle, had tremendous chi reserves in his body for his entire life, which he subtly sets up for the climactic fight scene by recovering from a serious stab wound, concussion and poisoning, in the span of about an hour, then casually mentioning that he's never had to go to a hospital in his life. Only after a near-death experience (that is to say, the Big Bad delivered sufficient damage that the surprisingly powerful Obi Wans had to mummify him) was he able to use it consciously.
Luke and Anakin Skywalker in their respective Star Wars trilogies. The Force was strong with those ones.
In the Tortall Universe books, there is a subversion on two levels. In the first, just about half the population has "The Gift", and it's actually called that. What it is is basically the ability to use magic - and that's it. Some of them ''do'' think they're better than everyone and two are the Big Bad of their series, but most are decent and nice. The second subversion is closer to the trope idea are wild magic; a much rarer 'gift' which has specific abilities. The only characters to have this (Daine and Tobe) are The Hero and a different hero's Side Kick.
Lord Voldemort of Harry Potter. From a very young age, Tom Marvolo Riddle was treated as an amazing prodigy and given special preference and instruction. Dumbledore likewise. Voldy just chose a very different path than Albus.
In The Wheel of Time series, some channelers require training and others have this, such as Nynaeve.
Subverted in that wilders (as 'gifted' channelers are known) are no more likely to turn evil than any others, they just have a harder time accessing their power before they undergo special training to break their 'block' that's sealing the power away.
Flipped in Thief of Time with Lobsang Ludd, who is naturally great at both the theory and practice of time manipulation. None of the teachers among the History Monks like him, because you can't teach someone who already knows everything.
River Tam of Firefly is a genius prodigy who can basically do anything she puts her mind to with incredible ease, and was already in some form of college by the time she was 14. As her brother Simon puts it, "River wasn't just gifted... she was a gift." After her Mind Rape, she is still extremely intelligent and talented, but it's often hidden behind many layers of insanity.
Sylar of Heroes has the power of "intuitive aptitude", and thus is able to rapidly master new superpowers over the course of a few days, whereas the people he takes them from tend to suffer from How Do I Shot Web? or Superpower Meltdown even after living with their powers for several months. Then again, he has to crack open people's skulls and take their brains to get the powers in the first place, so the "evil" part is kind of a chicken or the egg thing with him. Compare the heroic Peter Petrelli, who also can absorb powers (without stealing brains) but is pretty incompetent with them and needs to spend considerable time training to get them to work properly.
Troy Barnes from Community has the Gift— for plumbing and air-conditioner repair. Considering that in their world, air conditioning repairmen secretly control the world, its kind of a big deal.
Morgana Pendragon from Merlin When she finally accepts her magical powers, sometime between seasons three and four, she becomes a High Priestess after one year of training. She eventually becomes a serious threat to Merlin, who is the most powerful Warlock in history. She is a skilled swordswoman and is always able to get her hands on an army.
Ars Magica has this at the center of its premise. Magical ability (referred to as the gift) instantly puts the primary player character class, magi, above the rest of the mundane world. This is reasonable enough; its setting, Mythic Europe, is a world of medieval beliefs after all. However, while the magi of the Order of Hermes swear an oath not to do anything that deprives another magus of his right to arcane power, or brings the ire of the mundane authority on the Order as a whole, any given mage is perfectly within his or her rights to abuse anyone who isn't a mage at their leisure, including their own apprentices. A particular list of legal cases includes the case of one mage who tortured several of his apprentices to death and was found to have committed no crime.
As with the above, Unknown Armies has the gift of magic being incredibly rare. The thing is, it tends to cause Adepts to become... well, insane. The technical definition is that they become obsessed with things like taking risks, cutting themselves up, or saving money, to the exclusion of all else. Oh, and the only reason mages aren't ruling the world is because they're terrified of what could happen when the world finds out about magic. Otherwise, their obsessions and magic cause them to look down on normal people, referring to them as sheep.
Nearly every single player video game falls under this trope, as the main character (played by you) is in some way 'special'.
In Final Fantasy Tactics A2, Adelle is one of "The Gifted", which grants her such boons as an extremely extended lifespan, unique abilities and near-instant mastery of (non-combat) skills. While she never falls into outright evil (except for a brief moment when she is Brainwashed and Crazy), The Gift causes her a large amount of angst towards the fact that it makes her "different".
Knights of the Old Republic plays with the trope, but it still ends up used straight: the player character learns the ways of the Force very, very quickly, "learning in weeks what has taken others years". Later in the game, there turns out to be an explanation for that - but that explanation means that the player character is Revan, who before that revelation had been described as a bit of a prodigy, and rather powerful.
Tower of God - 25th Baam's incredible Shinsoo resistance and his huge talent for learning Shinsoo techniques. While it's true that he himself was very pure-hearted, his immense talent drew a lot of jealousy towards him and made some of his "friends" develop crushing inferiority complexes that led to soul crushing events.
Others whoever were awed by his power and decided to risk their lives to follow him (Koon, Rak, and everyone who survived the tests.). That was the original plan anyway.
Even after Baam's "death" the team still decided to stick together to honor his memory.
All the irregulars who ever entered the Tower are gifted in such a way that they are considered a combination of Humanoid Abomination and Person of Mass Destruction. They are considered so dangerous and so rare that an Irregular is thought by many as sign of great upheaval.
In the Dilbert animated series, he tells Dogbert that he has "The Knack" (for engineering.) He accidentally drinks his boss's coffee, gets 'management germs,' and loses it for an episode. Hilarity Ensues.
In Avatar: The Last Airbender Azula and Toph are both among the most naturally gifted/talented characters in the series. Azula was born as a child prodigy but Toph is blind and learned how to depend on other senses, which just made her an unnaturally good earth bender. The former is Daddy's Little Villain and the latter is a loner who requires an entire season to soften up to her True Companions. Katara is a subversion; she taught herself waterbending and became a master soon after finding a master to formally train her. Aang is an inversion. His gift means it's his job to save the world.
Twilight Sparkle in My Little Pony: Friendship Is Magic is a unicorn whose unique magical talent is magic. Princess Celestia — a millenia old Physical Goddess — says that Twilight has the greatest magical potential she has ever seen in a unicorn. In the pilot Twilight does come off as slightly arrogant and dismisses the future members of her True Companions as silly fillies who are getting in the way of her efforts to stop Nightmare Moon. Much of the series is devoted to Twilight coming out of her sheltered bookworm lifestyle. A few episodes also deconstruct The Gift by showing that Twilight can't always control her vast magical power.
About half of the Manes have some form of The Gift. Aside from Twilight, Rainbow Dash is such a skilled flyer that she smashed records left and right when she attended a training camp, and Fluttershy is so good with animals that she can actually talk to them. While the other Manes are skilled at their chosen professions, they aren't on a plateau like those three.
Bo Jackson was, and still is, considered to be one of the most talented athletes in the history of sports. He is the only athlete in history to become an All-Star in two professional American sports. The only person that beats his talent is...
...Jim Thorpe. He won Olympic gold medals in both the Pentathlon and Decathlon, and played at the professional level in Baseball, Basketball, and American Football.
Muhammad Ali was born with incredible speed, reflexes, and coordination. His coach trained him in an unorthodox way; instead of telling Ali what to do, Dundee instead just let him do what he felt was natural and simply 'suggested' things to do. Ali would simply do them.