Eat it, Tetris - Klax is the best puzzle game ever...
http://i42.photobucket.com/albums/e334/bVork/klax.jpg
Note that I'm about to score a lot of points.
Apart from the obvious fighters (SvC:MotM, Last Blade, and SS2) and the Metal Slug games, I love Magical Drop Pocket.
http://i42.photobucket.com/albums/e3...droppocket.jpg
On Gamegear, Crystal Warriors is my favourite game. I think it was the very first SRPG I played. The two Aleste games are great, but I never got to play those back in The Day.
http://i42.photobucket.com/albums/e3...alwarriors.jpg
The Gameboy (mono) has the much-maligned Pokemon series. I never want to see another friggin' Pokemon game again. Pokemon Blue ate up an obscene amount of my life... the current save has 80 hours on it, and that's the third playthrough. I know I spent even longer on the first two. I'd post a screenshot of the game, but that would require playing it. And I never want to touch another Pokemon game ever again.
Donkey Kong 94, LoZ: Link's Awakening, and Final Fantasy Adventure are my other favourites on the system. Others have already posted screenshots, so I feel no need to waste bandwidth and time.
I never got interested in GBC. I guess Metal Gear Solid is quite good, but the system never impressed me enough to get me to buy one - I was much more interested in the NGPC. And I never really got around to buying any GBC games once I bought a GBA.
I don't really consider the Turbo Express and Sega Nomad to be true handhelds, but I guess the same could be said for the Gamegear. Not owning the Turbo Express, I imagine that Devil's Crush on TG16 would be excellent as a handheld game. As for the Nomad, Mortal Kombat 2 seems to be the game that stays in the system the most. Streets of Rage 2 also sees a lot of play on it. Anything fast-moving like Sonic and Gunstar Heroes really suffers from the blurry LCD screen. I imagine Phantasy Star 2 (and 3 and 4...) would work quite well on the system, but I haven't gotten around to buying them yet.
