I feel like some of you look at the console as being too one-dimensional... Yeah, there are tons of shooters and that is the primary genre for the system, but there are lots of other types of games, too. Like I said, I know it plays shooters - what else? I collect anything fun to me and I try not to limit myself to one console or one type of game.
This is like saying the SNES and PlayStation only play JRPGs...
I don't think anyone is suggesting you only play shooters, or that those are the only games worth playing on the system...but to be honest, the library is lacking otherwise, especially with hucards. That's not to say that there aren't any great non-shooter games on the system, as there are a decent handful...but once you've got that handful of (non-shooter) games you're interested in, you're left with nothing but shmups. It's like eating all the chex & rye chips out of a bag of snack mix, and being left with nothing but a big sack of pretzels. Good thing a good many of those pretzels are delicious...
It's especially hard if you've got a PCE instead of a turbo, since games like Dungeon Explorer, Neutopia, etc. aren't quite as appealing when you can't read English. I really don't see the point in playing any RPG if you can't read the story, even if the story makes up only .001% of the game.
Instead of telling you my favorite games, I'll go through genres & list games I think are worthwhile. I'll try to keep it cheap, too.
Racing/Driving:
Chase H.Q. - chase down criminals & crash into them until they're immobilized. Plays similarly to Out Run. $15-$20
Power Drift - toony kart-racing game with interesting fake sprite scaling effects. $15-$25
$$$Out Run - if you're a die-hard fan of the game, it might be worthwhile to you, but I don't find it as good as the Genesis version & definitely not for the price it goes for, $45-$60
Platformers:
Legendary Axe I and II - sort of castlevania-esque gameplay...honestly I don't find the graphics appealing in either of these games, but the gameplay is solid. $20-$30
New Zealand Story - Cutesy, arcadey platformer where you play a Kiwi trying to rescue his fellows, and it's rife with Taito's usual eccentric charm. $20-$30
$$$Mizubaku Daibouken - if you like games like New Zealand Story, you'll probably like this one even more. Be prepared to shell out close to $100 for it, though.
Run & Guns:
Veigues Tactical Gladiator - forced-scrolling game where you pilot a mech & blow stuff up. The controls are clunky, but innovative; I personally believe they convey the feeling of operating a mech quite well). It's difficult & cheap at times, but the graphics and music are okay & you really can't beat $10-$15.
Genji Tsuushin Agedama - another mostly forced-scrolling run & gun themed from some manga/anime with which I'm unfamiliar, but the game is very bright & colorful and the gameplay is addicting to match. $20-$30
Genre-Defying Games:
Genpei Toumaden 1 and 2- I'm mentioning this because you have Dragon Spirit & I wasn't sure if you were fond of Namco, but these games are a lot of fun if you keep an open mind, especially Kannoni (the second one). They have a very strong mystical feudal Japan feeling & while the gameplay is downright punishing at times, I still feel they're worthwhile to play. Plus, they're next to dirt cheap at $10-$20 a pop.
Silent Debuggers - pseudo-3D corridor game where you sneak around & exterminate alien monsters on a space station. The gameplay & graphics might seem archaic by today's standards, but if you can get past that then this is one thrilling ride. Definitely worth $10-$15
$$$Gekibo: Gekisha Boy - Whacky game where you walk around taking pictures of crazy things as they appear, but this one is quite pricey. I've not done much shopping so I don't know how much it's going for, but I'm sure it's in excess of $100
Shooters you should play anyway:
After Burner: pseudo-3D game where you fly a jet and blow shit up. The PC Engine version is the best 16-bit console port, even better than the genesis version. Very fast and frantic, but you may only want to play in short bursts. $10-$15
Fantasy Zone - Another Sega arcade port, the bright colors and solid gameplay are retained even if the music takes a slight hit. For $5-$10, it's a must own.
I have yet to play any puzzle games or RPGs so I can't recommend any titles in those genres yet, but I think that list ought to keep you busy.