One of the things I love about the Dreamcast is the sheer youthful exuberance of it all. Most major games, even the less-than-perfect ones, feel fresh. There was this great atmosphere of creativity to Sega in its best years that set them apart from all other game companies, and the Dreamcast may be the system that burned brightest. I bought it about a year before the release of the PS2, and I had a sinking feeling as I walked out of Circuit City clutching my two boxes--a Dreamcast and Sonic Adventure--that the system was dying. I didn't care. I'm glad I didn't, because it was the last time I got to experience the pure joy and excitement of gaming on that scale.
Skies of Arcadia is by far my favorite Dreamcast game. It's also my all-time favorite RPG.
Shenmue and Sonic Adventure are good runners-up.
Shenmue is not really much of a game; most of the "gameplay" consists of doing mundane tasks, talking to poorly voice-acted NPCs, etc. But few games have felt so real to me, and for that reason, both Shenmue and its sequel are like nothing else out there.
Sonic Adventure is not widely appreciated. People love its sequel, but I actually like the first game more for its upbeat aesthetic and focus on fun gameplay (Sonic Adventure 2 spent too much time on gameplay styles that just weren't all that fun, while Sonic Adventure spends more time, relatively speaking, on speed-based levels and even the non speed-based levels are okay, excepting Big the Cat). The game is much less polished than it should be, but it brings as much of a smile to my face as very polished games like Mario Galaxy, so I can't find fault with it.
I also love Jet Grind Radio and Crazy Taxi.
It's difficult to overstate how important the Tony Hawk series is to me. I followed it from the first game through Underground 2 and loved most of my time with it. While I first played the original Tony Hawk on the N64, I bought it on the Dreamcast and put some serious hours into that version. It may be surprising, but I didn't even know what punk rock was before I first played Tony Hawk's Pro Skater. Hearing Goldfinger, New Girl, the less-censored version of Police Truck blew the 13-year-old version of me away. I became addicted to that kind of music. So mostly due to THPS, as far as I'm concerned, the Dreamcast ushered me out of innocent adolescence and into properly apathetic teenage years.
Basically, the Dreamcast was my go-to platform for everything after the N64, and even the non-exclusives feel like part of its ecosystem. Gaming has not been the same for me since this system's death.