Pikachu in Pokémon Yellow is unique among Pokémon in that it will never get in a Poké Ball. This is a trait that hasn't been shared with any other Pokémon in the main games, making it particularly stand out among the series. However, there is a fairly easy way to explain why this specific Pikachu refuses to enter a ball.
Pikachu is the mascot of the Pokémon franchise. It appears in almost every game associated with the series, and is the main character Ash's signature Pokémon in the anime. It is easily the most recognizable Pokémon in existence, and any reference to Pokémon will almost definitely involve Pikachu in some way. It is not a stretch to say that Pikachu is the world's most popular Pokémon. When Pokémon Yellow was released, not only did it update several elements of Red and Blue, but it also let players have a Pikachu as their starting Pokémon for the first time.
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In Pokémon Yellow, Pikachu will refuse to go into a Poké Ball, and will instead follow the player around on the map. This is actually one of the game's intended mechanics, as the player can talk to Pikachu at any time. Pikachu's reactions will show how they are feeling at the moment. At first, they will be quite cold towards the player, similar to how Pikachu was in the first episode of the anime. However, as time goes on, the player will be able to see Pikachu's reaction towards them change in what could be seen as a predecessor to Pokémon's hidden friendship stat. By the end of the game, Pikachu will likely be very happy with them. If Pikachu had stayed in its Poké Ball, then this mechanic couldn't have worked.
Pikachu remaining out of his Poké Ball could also easily be a
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