I like quite a few obscure games. It's from that time when you got games as a present from people that didn't know the first thing about gaming so you'd get the most interesting things. I'm glad it happened though, and I have quite a few of them in my list in the 50 Greatest Games Thread.
My favorites are;
Kula World, PS1
A Puzzle game where you're a ball rolling around all the sides of blocks and platforms trying to collect keys and reaching the end of the levels.
Asterix, PS1
A simple turn-based strategy game in the veign of Total War or the Risk Board-game, with a few platforming or beat-em-up mini-games thrown in.
Dynamite Cop, DC
An action beat-em-up game where every object in the area is a weapon. Choose from three different characters for three different paths to the end.
Step-On-It!, Mac
By far the most obscure game in this post, this is a puzzle game released only on Mac where the player must create and remove blocks and avoid enemies and hazards to get a key and exit the level. It's basically a clone (but with original levels and such) from the Arcade/NES game Solomon's Key.
