Fake Pokémon have been the subject of rumors since the franchise started in the mid-90s. From the fabled "PokéGods" to Pikablu to the fabled garden behind Bill's House, many children have spread rumors about artificial Pokémon they claim exist in the game.
One of the most common rumors has been glitch Pokémon MissingNo. can somehow transform into the fabled Mewthree. After all, Mew's existence is a complete surprise in-game, so why would it be so illogical for there to be an improved clone of Mewtwo in the game? Unlike the other Pokémon urban legends, Mewthree actually does exist -- from a certain point of view.
The Rumors
Rumors persisted in the original game of a group of entities known as the PokéGods -- super-legendaries that existed hidden in the game's code. These spread like wild-fire following the revelation that Mew, a 151st Pokémon, existed in the game. To make matters more complicated, Second Generation Pokémon like Togepi and Donphan began to emerge in both the anime and Japanese magazines, indicating that there were Pokémon beyond the initial 151 that existed. These seemed to validate confidence that the PokéGods existed -- Mewthree, an evolved form of Mewtwo, among them.
There was no real way to get these PokéGods, of course, but the rumors claimed you needed game-breaking cheat codes to hack into the game to get them, although these Action Replay and Game Shark codes often proved either ineffective or glitch creating. Other ways included getting to Bill's Hidden Garden (which doesn't exist) or beating the Pokémon League with a specific team of Pokémon. Obviously, none of these worked.
When scans of Pokémon: The First Movie came stateside, they seemed to reveal what many fans believed to be the rumored Pikablu (in actuality, Marill) as well as a Mewtwo-esque Pokémon fans believed to be Mewthree. In actuality, they saw Mewtwo wearing Giovanni's armor from the start of the film. Later on, some fans believed Lucario to be a new clone of Mew. An entire rom-hack of the third generation Pokémon games, Pokémon ChaosBlack, featured Mewthree as a fake Pokémon. Thanks to Mega-Evolutions, Game Freak managed to give fans Mewthree in the form of Mega Mewtwo X and Y, two new evolved forms of Mewtwo that proved super powerful.
The "Real" Mewthree
This real Mewthree exists in the Pokémon gag-manga, Pokémon Pocket Monsters. The main character, Red, travels not with a Pikachu as in the anime, but rather a Clefairy, one who not only has the ability to talk but also has a crass sense of humor. In the manga's twenty-fifth chapter, "Bring Down the Strongest Pokémon!!", Giovanni forces Bill and his wife to create Mewtwo. Ultimately, Mewtwo is freed from Giovanni's clutches, along with a Mew he snags.
During these events, Clefairy absorbs some of Mewtwo's DNA, becoming a Mewtwo-esque entity. At this point, Clefairy jokes, "This makes me Mewthree!" before transforming back to his normal Clefairy state. This gag is a one-off reference to the real Mewthree rumors that had been circulating around for years. When Mewtwo absorbs some of Clefairy's DNA, the DNA transfer process vastly increases the power of a Pokémon, making them temporarily stronger than either Pokémon are on their own. This means that Mewthree is stronger than Mewtwo -- but also that Mewtwo with Clefairy's DNA is comparably strong.
This is not the first time the Pokémon Pocket Monsters manga featured unconfirmed Pokémon in its pages. Brock's gym is home to many strange Pokémon that can't be identified, while many other Pokémon are off-model in the pages, such as a Diglett floating without ground around it and a three-eyed Persian. Regardless, this is the closest fans ever came to seeing a real Mewthree in any official Pokémon material.