Nanoha was originally conceived as the Token Mini-Moe of Triangle Heart 3: Sweet Songs Forever whose only role was to be the cute and innocent little sister of main protagonist Kyouya. Even in the mini-scenario where she became a Magical Girl, she was more adorable than threatening with her heart-shaped wand with little wings. Then when The Anime of the Game came, Nanoha's wand was replaced with a Swiss Army WeaponWave Motion Gun as she was re-imagined into a Little Miss Badass savior of The Multiverse who have been called a "(White) Devil" by foes and fans alike due to her ferocity and destructive powers which she does not hesitate to use on even her friends. Befriending, you know? The movie adaptations make her even more badass! In her first movie battle, she obliterate three monsters down at once with one attack. (she didn't learn Divine Buster until episode 3). Her Starlight Breaker can nuke a city now and her A.C.S. Driver can smash Reinforce through several skyscraper sized rocks!
In most of the video games, a Mewtwo is vastly superior to a Mew of the same Level in practically all areas; however, in Pokémon: The First Movie, Mew was powerful enough to fight Mewtwo to a standstill.
Roxie in Pokémon Black 2 and White 2: The second gym leader who isn't too hard to beat if you plan ahead. Roxie in the anime: The last Unova gym leader (well for Ash) who has Pokemon that are so strong, Ash has to use 6 Pokemon to defeat only 3 Pokemon.
In Pokémon Black and White, N, while still wanting to free Pokémon from their trainers, is a relatively tame person that releases the Pokémon he uses between battles with the player character. In Pokémon Best Wishes 2 Episode N, N is a much more active character that manages to fight off Team Rocket without using any Pokémon, only summoning some to heal Ash's Pikachu after the ordeal in his introductory episode.
Ash's Pikachu is extremely powerful, well beyond any average Pikachu (a fact that is pointed out early in the series). It can even knock out Ground-type Pokémon, when in the games Ground-types are immune to electric attacks.
The iconic red-and-white Poké Ball. In the games they're only really used very early into the game as they have a low catch rate. They're mostly only useful for very low-level Pokemon and even then it's not unheard of an under level 10 Pokemon to break out the Pokeball if their HP isn't down enough. In the animé, however, all the other Poké Balls (such as the Ultra Ball and Dusk Ball) are basically nonexistent and are little more than cameos most of the time. Basic Poké Balls are used to catch any Pokémon, no matter how powerful. It's extremely rare to see Pokemon not being caught in a basic Poké Ball.
The legendary birds in the games aren't said to be anything more than very rare and exceptionally powerful bird Pokémon. Pokémon 2000 makes them Physical Gods whose very existence controls the weather and are more than capable of ending it entirely. The three of them are even a match for Lugia, who's meant to quell their fighting.
Greninja is already among the most powerful starter Pokémon (especially if it has Protean), but Ash's Greninja outright gets a Super Mode exclusive to it that makes it even more powerful.
In the games, while Kyurem can be powerful on its own right, it pales compared to the rest of the trio master. And it needs to fuse with Zekrom or Reshiram with a DNA Splicer to become Black/White Kyurem. In Pokémon: Kyurem vs. The Sword of Justice, not only is Kyurem a dangerous and feared Pokémon by itself, but it can also transform without needing to absorb either of the two dragons.
Lillie. In her home game, she's an Actual Pacifist who hates Pokemon fighting and is only inspired to become a trainer because of how brave and strong the Player Character is. In the anime, she has a fear of Pokemon, but overcomes it and gets her first Pokemon, the Alolan Vulpix, who easily defeats Team Rocket.
Most of the Gym Leaders. In the games and anime, their job is to, well, job to the main characters. In this manga, they're so strong that it's generally accepted that only an extreme badass can beat them all note In this 'verse, you don't need any badges to actually compete in the League, but having all 8 badges mean you can skip the preliminaries. This incentive guarantees plenty of challengers. Furthermore, being a Gym Leader also means that your side job is to protect your region from things such as, you know, various terrorist groups. So yeah, expect to see a lot more ass-whooping coming from these guys.
In the games, Delibird is considered to be one of the weakest Pokémon as its overall stats are rather low, it cannot evolve and it only learns Present via Level Up. The Big Bad of the Gold, Silver, and Crystal arc uses one as his main Pokémon, being so well trained it can hold it's own against a Ho-Oh.
In /Pokémon Origins, Giovanni. In the games, while the Big Bad and a powerful opponent, his Pokemon for the Gym Battle (the strongest team he fields) are either inferior or on par with Lorelei's, the weakest of the Elite Four. Here he's far stronger, on par or stronger than even Blue's final team, using two Pokemon to Red's six in the Gym Battle and his Rhyhorn alone beating five of Red's before suffering a Double KO (and could have outright won had it not suffered recoil damage from two Take Downs) and Charizard barely winning against Rhydon.
In the Mitsuki Oosawa manga adaptation of Fire Emblem: Genealogy of the Holy War, White Magician Girl Deirdre moves quite closer to the Competence Zone. Especially when she takes King Clement's castle with only back up from Raquesis and Aya, without killing anyone and relaying in trickery and magic than brute force, the expanded version of her confrontation with Manfroy in which she does get kidnapped to follow the melodramatic Downer Ending of the first generation but still attempts to fight him in self-defense and to protect little Shanan (plus she may have been captured but succeeds in protecting the castle and everyone else in it); and the rendition of her own death, where she had already guessed that Julius would go Enfante Terrible on her so she sends Julia away earlier and protects Arvis from Julius.
Several characters who suffer from a bad case of What Measure Is a Non-Badass? in the Naruto manga finally get to catch a break in the anime. By far the most glaring example is Hinata, who got to learn a powerful technique the anime team just made up for her, defeat menacing Filler Villains on her own and actually land a hit during what was in the manga a Curb-Stomp Battle (in the process extrapolating what her Gentle Step Twin Lion Fists actually does, since in the manga she was knocked out before getting a chance to do more than activate it).
Syaoran Li to an extent in the Cardcaptor Sakura anime. While still under the same inability to catch cards himself, stipulations allow him to earn several cards of his own, most of which he is rather adept with. This allows him to act as a far more effective rival for Sakura, even earning his own trial in the Final Judgement. Even after losing all his cards to Sakura, his acts in assisting and protecting her often prove astonishing, with him suggested to still be at a similar or higher level of magical power than her by the closing movie (albeit partly due to lacking a plot point of the manga that rendered Sakura unmatchable in power).
In the original Cutey Honey series, Natsuko was mostly a Distressed Damsel and Honey's friend/potential lesbian lover. In the live-action movie and the RE Cutie Honey OVA series, she's a badass adult policewoman with a talent for using firearms (and still is Honey's potential lesbian lover).
In Mai-HiME and Mai-Otome, a few characters' power levels vary based on which series and media they appear in.
Shizuru typically has the superpowers associated with the series in all versions except the Mai-HiME manga.
Haruka is a Badass Normal in the Mai-HiME anime who can't win any fights but lives long enough after Yukino's child is killed to headbutt Shizuru and toss Yukino her armband, but is a Hime in the manga. The opposite is true in Mai-Otome; she is a powerful Otome in the anime, endowed with Super Strength and second in her class after Shizuru, but an ordinary police officer in the manga.
Both of Yukino's Mai-HiME incarnations are Himes, but play supporting roles (in the anime, she primarily uses her powers for surveillance, while in the manga, she uses hers to support Haruka). In Mai-Otome, she has no powers, and is the president of Airies in the anime and a Windbloom police officer in the manga (albeit skilled enough with a sniper rifle to arguably be more effective in battle than her Hime incarnations)
Persona 4: The Animation gives has Izanagi. In the game, Izanagi's just a cool-looking Persona with low-tier stat growth/elemental affinities/skills that most players will have gotten rid of by the second dungeon; but in the anime he comes off as the biggest badass of the entire bunch, regularly being the go to Persona when Yu needs to wipe out a particularly dangerous Shadow. Then again, it is possible to fuse a really badass Izanagi with skills like Primal Force, Angelic Grace, and Power Charge. Much later in the game, that is.
Similarly, Beelzebub is a good, late-game Persona, but far from the best. In the anime, Yu fuses him during the battle with Shadow Naoto, uses Megidolaon, and turns the entire dungeon into a smoking crater.
Adachi. In the game he was kind of a pushover, however in the anime he manages to thrash Yu quite a bit, and is in control of three Reapers to kick the crap out of the rest of the protagonists (The Reaper being a fairly powerful optional Boss in the game).
Persona 4 Golden: The Animation takes this to ridiculous levels, with Izanagi going One-Man Army on thousands of shadows the first time it's summoned; using late game moves such as Null Physical and Maziodyne. Magatsu Izanagi is, likewise, much more powerful. Adachi uses it to defeat every persona that Yu summons (all of them really powerful, late-game personas) before it's finally stopped with a Suicide Charge by Izanagi, forcing Yu to resort to hisbare hands to finish the fight. and afterwards, under Yu's control, it takes out Ameno-Sagiri with one slice.
The Digimon franchise has Gotsumon. Several series, which are AU to each other, have the little guy as a Recurrer with the same or similar personality and voice portrayal. Each version tends to be badder than the last. Digimon Adventure? A cute little guy who'd rather party than fight, his death (along with that of his partner, Pumpkinmon) was used to show how much of a Bad Boss the arc's villain was. Digimon Frontier? His tiny little pebble attack has grown into the ability to create giant boulders and he is able to hold off enemy enforcers who, according to their stats (type, level, element, etc.), should stomp him like a bug without even knowing they'd done so. He can now become Meteormon, a Palette Swap who surprises everyone by being ultra-powerful. Digimon Savers/Digimon Data Squad? let's make him bad again, but good at it! He bedevils the team for the entirety of their first extended stay in the digital world, then becomes Meteormon. They laugh at his dramatic, Large Ham-moment-prefaced transformation into himself, but with lighter coloring for all of two seconds before he starts wiping the floor with them.
Wargreymon began its life as a Power-Up Letdown for Metalgreymon, and was widely reviled for being relatively weak for it's level, having the lowest possible lifespan and even having an abnormally high after battle injury rate. Overtime, as a result of animated adaptations, his flaws tend to be glossed over, lending him a needed air of competence.
Rune Soldier Louie: Jeanie's fighting ability changes drastically between whether you're reading the manga, or watching the anime. In the manga version, Louie lays her outwith one punch, when they first meet in chapter 1. Whereas in the anime, she's a much better fighter. When she and Louie fight in episode 12, not only does she outlast him, she knocks him out (seen from 13:23-16:57).
While Vegito was already a Story-Breaker Power in the Dragon Ball manga, the Dragon Ball Z anime made him even more so. In the manga, he immediately turns Super Saiyan when fighting Majin Buu, and flattens him. In the anime, he spends an episode fighting Buu without turning Super Saiyan, which as onesided as the fight after he transforms, he still dominates the fight to the point it's clear he didn't actually need to transform to overpower Buu, he just did to humiliate him.
A Downplayed example, but Freeza put up a much better fight with Super Saiyan Goku in the anime compared to the manga. In the manga, although Freeza did decently, it was clear from the outset that he never had a chance of winning and that was before his power started to fail him. In the anime, Freeza got several good hits in and gave Goku the fight of his life (although he still had no real chance of winning) before his stamina gave and was reduced to a non-threat by the end of the battle.
Freeza's return as Mecha Freeza. In the manga Trunks takes Freeza out while he was distracted, not giving him the chance to power-up. It was hinted that Trunks was only as strong as Super Saiyan Goku on Namek. In the anime, Trunks took Freeza apart even when he dropped a Death Ball on Trunks' head, trying to destroy the planet.
Yamcha, Tenshinhan and Chiaotzu got the chance to defeat the Ginyu Force in two filler episodes, Yamcha defeating Recoome, the one who just beat the crap out of Vegeta, Gohan and Krillin not long before. In another filler scene, Yamcha defeated Ulibu, a character who is probably stronger than Freeza. That's right, the Memetic Loser defeated a guy who is stronger than Freeza.
In the manga, Dabura's skills and abilities are extremely lacking in comparison to his anime counterpart. He mostly uses sneak attacks and in fact he never once throws a single physical punch in the entire manga. He's unable to overpower a rusty SSJ Gohan and is quickly dealt with by Buu before he can even throw a single attack. The anime, however, amps him up, allowing him to hold his own against Gohan and even standing up to Buu for quite some time.
While Goku is never a wimp in any media, The Path To Power amps him up even more. Case in point: in the original manga and anime, his first attempt at the Kamehameha Wave only results in a small energy beam that's only strong enough to destroy Bulma's car, while in The Path to Power, said first attempt results in a Wave Motion Gun that completely obliterates a good chunk of the Red Ribbon Army's armada.
Android 8 is quite powerful in the series proper (stronger than Goku at the point they met), but his strength is Up to Eleven in the movie Path to Power. He is able to fly, dash, and nearly kills Goku, who is also made stronger. He also fights a mecha that knocked Goku unconscious for several minutes and pushes it back.
The Super Saiyan God form gets this treatment in Dragon Ball Super. In Dragon Ball Z: Battle of Gods, Super Saiyan God was the definition of Awesome, but Impractical: it gave Goku the power to battle Beerus on more or less equal grounds, but it only lasted for a few minutes — short enough that Goku lost the transformation in the middle of the fight, and actually performed better fighting as a normal Super Saiyan, since he absorbed the power of godhood. Here, while the end result is pretty much the same, Super Saiyan God lasts much longer and overall has a better showing, including slowly destroying the universe just by fighting Beerus and, even if it was brief, proves able to stand up to Beerus' full power. In the movie, by comparison, Beerus never used more than 70% of his power. It also seems to give Goku a type of Healing Factor, since he recovers from being impaled through the stomach.
Gohan's Bad Future counterpart gets this treatment as well. In the manga story of Trunks' timeline, Gohan is killed by Android 17, who claimed to have not even used half of his full power to do so. In the anime adaptation of The History of Trunks, Gohan is actually stronger than both 17 and 18 individually, only losing because the two ganged up on him.
Frieza himself also gets this treatment in Dragon Ball Super. In the original Resurrection 'F', Final Form Frieza is completely dominated by Base Goku and, as Golden Frieza, he runs out of energy too fast to really give the impression he could have beaten Goku. In Super, Final Form Frieza matches Base Goku in terms of speed and power, and Golden Frieza beats SSGSS Goku around like a ragdoll, with Goku nearly losing consciousness as a result.
Botamo in both the manga and anime versions of Super is such a Stone Wall that not even Goku can harm him. However, in the manga, Goku's attacks could send him flying, while in the anime, Goku's attacks couldn't even budge him.
This trope is invoked for Yamcha in the Dragon Ball side manga, The Case of Being Reincarnated as Yamcha. As the title suggests, the main character is an ordinary teenager who dies and gets reincarnated as Yamcha. Being well aware of what happens to Yamcha, he decides to train hard and change that. And he succeeds. By the time the Saiyans arrive, his power level is over 10,000, letting him easily wipe out all the Saibamen singlehanded and defeat Nappa on his own. Even though he's not as strong as Vegeta, he's still powerful enough to team up with Goku to defeat him.
Shimoneta: Despite being the manga's main antagonist, White Peak ends up being a pushover when SOX confronts him at the end of chapter 12. Ayame and Anna drop him with a single kick that knocks him unconscious. The anime changes things by giving him a full-length fight scene, in which, it takes Ayame, Anna, Tanukichi, and Oboro to finally bring him down.
The Sonic X anime did this with Shadow the Hedgehog. He's already incredibly powerful in the video game series, but not terribly so. In the anime? He's practically The Juggernaut in that very few characters are even capable of slowing him down, even Sonic himself. He also spends most of the third season as a One-Man Army against the mooks of that season's Big Bad.
Fate/stay night [Unlimited Blade Works] does this to Berserker. In the original Fate/stay night, he fights like, well, a mindless berserker. In Unlimited Blade Works, he demonstrates swordsmanship skill and awareness of his surroundings, like when Saber traps his sword and attempts to decapitate him, only for him to let go of his sword and backflip out of the way.