In most of the video games, a Mewtwo is vastly superior to a Mew of the same Level in practically all areas; however, in Pokemon The First Movie, Mew was powerful enough to fight Mewtwo to a standstill. (This was also the case with Mons like Celebi and Jirachi in their anime appearances, much tougher than they were in the games.)
Most of the Gym Leaders in Pokémon Special. They're so strong that it's generally accepted that only an extreme Bad Ass can beat them allnote 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.
Delibird is one of the weakest Pokémon stat-wise, and is considered a joke to be used in competitive battle. The Big Bad of the Gold, Silver, and Crystal arc uses it as his main Pokémon, being so well trained it can hold it's own against Ho-Oh.
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.
In the Mitsuki Oosawa manga adaptation of Fire Emblem Seisen no Keifu, White Magician Girl Diadora moves closer to the Competence Zone. Especially when she takes King Clement's castle with only back up from Lachesis and Aira, without killing anyone and relaying in trickery and magic than brute force, and 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.
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.
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 apt 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 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 us 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 turn 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 Down Played example, but Frieza put up a much better fight with Super Saiyan Goku in the anime compared to the manga. In the manga, although Frieza did decently, it was clear from the onsite that he never had a chance of winning and that was before his power started to fail him. In the anime, Frieza 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.
Frieza's return as Mecha Frieza. In the manga Trunks takes Frieza 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 Frieza apart even when he dropped a Death Ball on Trunks' head, trying to destroy the planet.
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