Go places and do things
I think my favorite game genre is “go places and do things”. Especially when there are multiple objectives that I can pursue in an order and pace of my choosing.
Prominent examples include 3D platformers like Jak & Daxter and Ratchet & Clank, open world games like inFAMOUS and Saints Row, and action RPGs like Dragon Quest Heroes II and Kingdoms of Amalur: Reckoning.
Some games from adjacent genres qualify as well, such as MMO City of Heroes, Metroidvania SteamWorld Dig 2, and life sim Disney Magical World.