Super Mario Galaxy

Acquire 121st Star:
First acquire all 120 Stars. This gets you Luigi. Finish the whole game with Luigi, beating Bowser at the end. After this, a new galaxy can be had. Beat this final galaxy to get the 121st Star.

Acquire Luigi:
Acquire all 120 Stars in the game; next, go to Rosalina, battling Bowser one last time. After the credits a message will display telling you Luigi is a playable character.

Unlimited 1-Ups:
If you acquire any 1-Up Mushroom at the Comet Observatory, you can then navigate into any dome, head out, and arrive back at the Comet Observatory without having played any stage.