Where is the team rocket boss go after you beat him at game corner?

The Team Rocket Boss, Giovanni, doesn't actually appear at the Game Corner in any mainline Pokémon games.

He is the leader of Team Rocket in the following games:

* Pokémon Red, Blue, and Yellow: You face him in the Viridian City Gym.

* Pokémon Gold, Silver, and Crystal: You face him in the Silph Co. building.

* Pokémon Ruby, Sapphire, and Emerald: He is the leader of Team Aqua or Team Magma, depending on the game, and you face him after defeating the 7th gym leader.

* Pokémon FireRed and LeafGreen: Similar to the original games, he is the leader of Team Rocket and you face him in the Viridian City Gym.

There is no instance where you face Giovanni at a Game Corner.