Up to 100 friends, acquaintances or favorites.
During the Pokémon X and Y roundtable event the Player Search System was revealed. Acting as the game's multiplayer hub, it will show friends, acquaintances, and people passed on the bottom screen. Ideally this would be always connected to keep you up to connected to friends and other players.
Whenever you interact with a player they can be registered as an acquaintance, and after more than one interaction the game even prompts you to do so.
Up to 100 friends/favorites can be saved in the PSS. Friends will be automatically imported from your 3DS friends lists and added to the PSS