While not necessarily a Genesis fanboy. It was the only 16 bit console I owned. I borrowed a friends SNES to play Final Fantasy. But! There were some terrific titles for the Genesis that just don't have counterparts on other systems (that I'm aware of). Personal favorites are marked with a *. I'll leave out any arcade remakes, or games that you will find on other systems (except Dragon's Fury, I think the Genesis version is better).
Platformers:
All the Sonic the Hedgehog games are of course very good.
* Blaster Master 2 - If you played the first one on the NES it's more of the same, but bigger, better, bad-ass-ier, and with a really good soundtrack. I highly recommend it.
Dark Castle - Not really a platformer, but it's one of the hardest puzzle platformer games I have ever played. It's only 4 or 5 rooms, but it's the most dastardly 4 or 5 rooms you're likely to ever face... thanks to the sluggish controls.
* Gunstar Heroes - quite possibly one of the very best platform shooters ever made! Extra good fun when played with a friend.
Quack Shot: Starring Donald Duck - Yep it's a quirky little platformer, and I like it (So does Evil Dragon). It's short, it's fun. and it's addictive.
Ranger X - Similar to Turrican and Arrow Flash - the thing here is you kind of have to control yourself and a motorcycle thing at the same time. It's different, and hard to get the hang of, but once you do ... oh buddy... it's fun.
Revenge of Shinobi - It's a Sega Classic for a reason. Sure the controls are a bit cumbersome, but it's stunning and gorgeous and very satisfying to beat.
World of Illusion: Starring Micky Mouse and Donald Duck - The graphics are stunning, the music is timeless, and the gameplay is almost flawless. It's fun and not too hard. Playing two player is fun as well.
RPGs:
Beyond Oasis is a very good Action RPG similar to The Secret of Mana. I don't remember finding it very challenging, but it was pretty fun.
Shining in the Darkness - A first person rougelike set in the Shining Force universe. It's a very long game, a bit dull in parts, but still a good addition to any collection. Save states are a must though.
* Shining Force I and II: I love the turn based strategy of these games, and really like the "promotion" system. So many hours... so... many... hours...
Beat-em Ups: (Please note I am not a big fan of the genre)
Cyborg Justice - I can already hear the moaning. This game is TERRIBLE! It very well may be. For me it's a guilty pleasure. If you take a little bit of time to learn the game mechanic it's a sinfully gratifying way to dismember everything in your path. Rip off an arm, toss it, pummel into a pile of scrap, kick him when he's down, reassemble him, rip off his torso and devour it's essence. Ahhh... so fun.
Shooters:
* M.U.S.H.A. - This is my absolute Genesis title - It's a vertical shooter whose level of difficulty is damn near perfect, Challenging, but not too much so. The music is phenomenal (if you like metal/hard rock) and the game play is smooth and deadly.
Arrow Flash - Is a unique and challenging shooter, but you can change forms to assist in navigating the terrain. I don't think I ever actually beat it, but I stands out because of it's interesting mechanic.
Target Earth - A tough, in your face, side-scrolling shooter. It's hard to describe exactly since you spend so much time walking on the ground, but there is some zero-g combat in there too. Very hard. There's a cool cheat though, if a friend starts using controller 2, they can take over enemy units and fight along side you.
Whiprush 2097 - It's tough, but not because of the bullets or enemies. It really goes out of it's way to smash your into things. I thought it was a refreshing change of pace for a shooter. Oh, and it looks nice too.
Racing:
* Road Rash I, II, and III - I love these games. Simple, aggressive, adrenaline mayhem on crowded roads with crowbars and chains. Elegant. My personal favorite is Road Rash II. My advice is to turn off the music, and blast some metal while playing, it's a much better ride.
Other:
Dragon's Fury - aka Devil's Crush - video pinball with a cool twist, there are bonus pinball mini-games that unlock stuff. My buddy and I played this until our eyes dried in their sockets. Quite fun, if you like pinball.
Dragon's Revenge - sequel to Dragon's Fury. Not as good IMO, but still worth a play if you like pinball.
* Starflight - If someone held a gun to my head, and told me to choose which game to never play again, this or M.U.S.H.A. I'd tell them to just shoot me. Starflight is a fantastic space exploration game with a real plot, and a fair amount of depth. So many people give up on this game too quickly. Give it a fair chance, and it very well may become one of your favorites too. I've replayed this game dozens of times.
Edit: World of Illusion is NOT a shooter.