First I want to do an honorable mention, that being Glover. You play as a magical glove brought to life and have to transport a ball that you can magically change the shape of through a level. Oddly kind of a fun game. This is a weird game, but I actually think its decently well known.
And the weirdest game I have ever played, Mort the Chicken for the PS1. You play as a chicken and you try to save chicks from alien creatures using things chickens do, such as pecking and gliding. Oh yeah the alien creatures are cubes. Just cubes. Different ones are different colors, but they are all just cubes. Its a really weird game.