Two immediately come to mind--the first being True Crime: NYC, a decent (in my opinion) GTA clone with fun, if glitchy, gameplay. You're basically a cop--good or bad, you choose--patrolling a decent-looking and accurate (at the time) Manhattan, stopping random street crime as well as going after bigger fish. I love the excitement of a high speed car chase with criminals through Times Square, I got a kick out of the many buildings you enter in pursuit of miscreants, stopping everything from attempted rapes to (rare) terrorist attacks on the Lincoln Tunnel. The gunpplay is fun, if short, but the main enjoyment I got from this game was the accurate layout of the city and its look, trash blowing freely though grimy streets (until you clean up crime in a particular area, changing its look somewhat), and the satisfaction of bringing down some criminal who tried to run.
the second is Space Channel 5 Part 2--I'm not alone here, but I may possibly be the only dude in the world who loves this game. The end of stage 2 is my favorite boss fight of all time. Just a great, fun game with a fantastic soundtrack. It's not for everyone...you basically dance your way to victory, amassing a crew of people to dance with you as you make your way though the missions.
What games do you play that you're sure you're the only one playing?