I'll list a few:
BioHazard Battle - Shooter where you are shooting at big bugs/monsters. Really like this game and don't believe it is on anything else
X-Men - Believe this was also only on the Genesis. Side scrolling beat-em up with X-Men characters. You get to play as NightCrawler and try to do his teleport, so that is my main reason for choosing it. Pretty fun if you don't mind 'not best ever' games.
Alex Kidd - Not a good game at all, but holds a special place in my heart. You get to punch dudes with an oversized fist, one-hit kills, and bikes and cars that jump. Also, you fight bosses through Ro Sham Bo.
Quackshot - As mentioned above. Donald with a plunger gun. Makes for an interesting platformer since you can 'make' platforms.
Vectorman - An alright action game. Kind of what Sega did in response to DKC, graphics were the main selling point if I remember. Game is fun though.
Thunderforce II - An overhead shooter , kind of gives you the whole map to go in, but there are borders that if you cross hurt you. Never played three, so don't know how that compares.
Herzog Zwei - A pre RTS, RTS. You have a main base and are constantly collecting currency. You play a jet fighter that can turn into a dude or a transport ship. You have to try to take out the opposing base by building troops tanks and stuff, most have at least two options. while also protecting your base. There is a score system in there as well. Never played two player, but hope to sometime. Takes a little bit to understand the game as well.
Decapattack - A platformer where you get to throw your headat bad guys.
Pretty sure you can get any of these for less than $20.
