Video games that let you play other video games, inside the game (gameception??)

[DragonMals]

The Original Animal Crossing allowed you to play NES games.
Metroid: Zero Mission allowed you to play a port of NEStroid.
If you find the secret area in Celeste, you can play Celeste Classic.
Puyo Puyo Box contained ports of the original Puyo Puyo and Puyo Puyo Tsu for the Arcade/Genesis.
The Legend of Dark Witch Renovation allows you to change the style of the gsme after unlocking the old style which perfectly replicates the Original 2014 experience. this is due to both games utilizing the same engine as each other.

 

[Trix_]

It's been a while, but I'm pretty sure that Always Sometimes Monsters had an arcade and you were required to play a game to pass a story sequence.

Hypnospace Outlaw also has the Enforcer minigame.
(I fucking love Hypnospace Outlaw and I can't wait for Dreamsettler).