I love strange games, so I have a couple honorable mentions first:
Whiplash - A genetically altered weasel breaks free of his cage and proceeds to wreak havok by using the genetically altered rabbit that's chained to his leg as a weapon.
Seaman - Raise a grotesque half-man/half-fish creature, have strange conversations with it and get constantly berated by it.
Y'know, as much as I wanted to dig up some little-known game to be the weirdest thing in my collection, it's one that I'd wager most people here are very familiar with. That doesn't make it any less weird tho:
Katamari Damacy - On a druken bender, your dad - the King of All Cosmos - destroys all of the stars. You have to recreate them by rolling random objects, food, animals, cars, houses, etc. into a giant ball. Then your dad tells you how poorly you did, with a voice that's literally a record scratching. It was weird about 10 years ago when I first played it, and it doesn't sound any less insane now.
