From Bulbapedia, the community-driven Pokémon encyclopedia.
A living Pokédex is a fan term for a game which not only has a complete Pokédex, but has one of every available species of Pokémon stored in its PC boxes as well, usually in their National Pokédex order. It follows the most literal interpretation of the slogan of the series, "Gotta catch 'em all!".
Perhaps one of the most challenging questions is what to do with the living Pokédex once it is complete. In particular, in this instance, or with any large Pokémon collection in general, the question of storage arises.
In Generation I and Generation II, the Pokémon Stadium games could be used to store large amounts of Pokémon for the purpose of a living Pokédex.
In Generation III, Pokémon Box was introduced, which allowed players to store their large collections without worrying about taking up PC space; in fact, players were rewarded for transferring more Pokémon into Pokémon Box, making the living Pokédex goal more appealing.
In Generation IV, players can use My Pokémon Ranch to store up to 1000 Pokémon, and up to 1500 with the updated version. Unlike Pokémon Stadium games and Pokémon Box, players could not use My Pokémon Ranch to transfer large amounts of Pokémon from one game to another very quickly.
In Generation V, there was no storage "game" available. However, trading between boxes could facilitate more trades per session. Also, beginning with Pokémon Black 2 and White 2, players could obtain additional rewards for completing the Pokédex.
In Generation VI and Generation VII, Pokémon Bank can store up to 3000 Pokémon online. The ability to transfer many Pokémon between games was restored.
In Generation VIII, Pokémon HOME can store up to 30 Pokémon online for free and up to 6,000 Pokémon online on a premium plan. On launch, this was the first instance of a game giving out a distinct reward for completing the Living Pokédex, with an Original Color Magearna. However, an update one month later reduced the reward's requirements to just completing the National Pokédex.