I know you've already gotten quite a few, but I'm throwing my hat in anyway:
Streets of Rage - A Beat-'em up in the style of Final Fight for the SNES, only way better and with a timeless soundtrack. Awesome with 2P.
Streets of Rage 2 - A Beat-'em up which took the original and added arcade-style graphics and a larger roster of characters with varying attributes. The best 2P game on the system.
Golden Axe - The classic arcade-style hack 'n' slash. Great with 2P.
The Revenge of Shinobi/The Super Shinobi - An excellent action platformer, with pixel-perfect precision. Quite a punishing game.
Shinobi III - Action platformer. easier than the prequel, but more cinematic with a more varied and fluid moveset. An epic game.
Space Harrier II - Third Person Rail Shooter. Not as good as the arcade original, but pretty damn good still. Challenging.
Comix Zone - A unique Beat-'em up game, where you have to jump between the pages of a comic to fight the bad guys. Awesome music and good combo-system make this a really satisfying game.
Super Thunderblade - Third Person Rail Shooter but with a helicopter twist, bitches. Hard as nails and best with a joystick.
Super Hang-On - An Arcade Racing Game, quite like OutRun or Rad Racer for the NES. A fine classic.
Dune II - If you've ever played Command & Conquer, then you'll recognize this. The daddy of the modern RTS, this version actually improved somewhat on the PC original with certain features that have been RTS staples ever since. Wildly addictive and hugely time-consuming.