Sega Genesis/Mega Drive for you foreigners

[Girl Drink Drunk]

Sidescrollers:
Shinobi III
Bonanza Bros.
Sonic the Hedgehog 3
Toe Jam and Earl: Panic on Funkotron
Jurrasic Park
Robocop Vrs the Terminator
Alladin
Rocket Knight Adventures and Sparkster
Spider-Man vrs the King Pin
Moonwalker
World of Illusion
Ristar
X-Men 2
B.O.B
Ghostbusters
Castlevania Bloodlines

Puzzle/Adventure:
Lemmings
Dune
Out of this World

Beat em ups:
Streets of Rage 2
TMNT: Hyperstone Heist
Battle Toads
Splatterhouse 2 & 3

RPGs:
Shining Force

Other:
Lethal Enforcers
Zero Tolerance
Truxton
Worms

Games like Dune, Out of this World, Zero Tolerance, and Lethal Enforcers may or may not be your thing so you might want to a few out on an emulator first, but otherwise, you'll probably like the other mentioned games.