I played a little of these before. I was extremely saddened by Jet Grind Radio's limited downloadable Graffiti due to the Dreamcast's dialup connection and long dead host site. It manual mentions using Jpeg images, but the only non-hacky way of transferring them was the Internet selection.
I played some Resident Evil Directors Cut, but I forgot about how easily games crashed back then due the most minor scratches you can't even see. I also played some Marvel vs Capcom. Stopped though because it wasn't the one I was thinking of.
But my best experience was popping in Vigilante 8: 2nd Offense. The whole idea of the bad guy basing his domination on vehicles essentially cars that float cracks me up. You can see just how silly it is in this video:
http://www.youtube.com/watch?v=9nGeB7lpiJ0Seriously, somebody watch that video, and tell me that isn't ridiculous? I read a little of the backstory thing in the game, and it says that we defeated them in 1974 or something. Then, by 2016 (that future), the bad guy is finally about to be put away for good. Be finds out about new time technology and steals it, then sends people to the past to stop the Vigilantes and create an empire for him...
It's ridiculously hilariously awesome.