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Aladdin, Toki, Mega Turican, Soldiers of Fortune (Chaos Engine), Kid Chamelon, Wonderboy in Monster Land, Fire Shark.

Congrats!! There are plenty of great games to play. I highly recommend as others have mentioned Vectorman, Gunstar Heroes, Golden Axe, Streets of Rage, Altered Beast, and Phantasy Star and Shining series if into RPG games. There are too many titles to list..

Let's see, I'll document what I think are the must-owns that have aged well (excluding the Sonic series, as you requested):

Gunstar Heroes
Shinobi III
Phantasy Star IV
Shining Force II
Streets of Rage 2
Thunder Force IV
ToeJam & Earl (I prefer Panic on Funkotron, but I'm in the minority)
Ecco the Dolphin (I prefer Tides of Time, but again - the minority)
Comix Zone
Ristar
Earthworm Jim
Rocket Knight Adventures
Vectorman

There are tons of other classics, but these are the ones that are relatively affordable that you simply must own. There are some other great titles that I feel either didn't age that well (Golden Axe, Strider, Castle of Illusion, etc.), have significantly superior contemporaries on the SNES (Castlevania: Bloodlines, TMNT: Hyperstone Heist, etc.), or are just too expensive to recommend (M.U.S.H.A., Pulseman, Alien Soldier, Panorama Cotton, GleyLancer, Eliminate Down, Yu Yu Hakusho, Herzog Zwei, Super Fantasy Zone, Twinkle Tale, and Monster World IV are all top-recommendations, but just cost too much to recommend) and then there are a few which I love, but people seem to have varying opinions of (Beyond Oasis, Ranger-X, Contra: Hard Corps, Dynamite Headdy, Sonic Spinball, etc.).

Get Zombies Ate my Neighbors, its not an exclusive but its awesome!

Contra: Hard Corps, Thunderforce III, and Rocket knight adventures!

One of my personal favorites is The Adventures of Batman and Robin

i second beyond oasis, wonderboy in monsterworld, and will add landstalker.

Thanks for all the feedback, I'm going to research some of these titles I haven't played and add some to my wishlist.

I'm jumping in a little late on this, but here are a few I don't think have been mentioned yet:

Castlevania Bloodlines
Desert Strike
Dragon's Fury
Ranger X
Strider

It may be a little overwhelming, but my entire greatest games list can be found here:

http://timewarpgamer.com/top_games/gen.html

NBA Jam
Mega Man: Wily Wars
Outrunners