Monster Party for the NES. It was bizarre, a bit disturbing, and thoroughly awesome.

Some PS1 games I remember being awesome but don't get much discussion 'round here:

Rascal
Jersey Devil
Medieval
Intelligent Cube (or was it Qube? cba to look it up)
Treasures of the Deep
Brave Fencer Musashi
Croc: Legend of the Gobbos
Mr. Driller