After being registered for 1 year, you will receive a notification to claim your Anniversary Gift. You have the option between two different variants of the current Anniversary Pokémon. Each year, a user will only be able to select one and will have to wait until their next anniversary to be able to choose the other.
Starting from PokéHeroes's first anniversary in 2014, you were able to claim an Anniversary Cake.
When claiming the gift, you will receive it directly in form of an egg.
Note: Hatching the egg will not count towards your chain.