Well, I’m not addicted to a single game so much as a genre of games. Specifically, chess. I’ve got three different chess games, plus two online chess clients I use to connect to Freechess.org (at least until I decide between them). I probably play Chessmaster the most.
I was never much into games until a friend of mine got me hooked on Sims. Now I have all the expansions–except for Sims Vacation. My little Gateway is barely out of toddlerhood but he’s doing funky things so I haven’t played much lately 
Subspace (now called Continuum). Massivly multi-player online space battle/flagging and teaming top-down Asteroids-style game that easy to learn and hard to master…
Currently alternating between Diablo II and Vampire Chronicles (on the Dreamcast). I expect in a few weeks it’ll be Morrowind (and nothing else), assuming my system can handle it.
The Administrative Assistant Funpack
Games for the Socially Challenged
Oh, whoops, you meant computer games. 
I’m also in classic game mode, like Quake II and its progeny and the occasional MAME marathon session. I expect to be addicted to Grand Theft Auto III if it ever comes out for PC.
Civ III, looks like I’m in good company here Also Operation Flashpoint. Difficult but thoroughly enjoyable.
I play lots of Diablo II expansion. Whenever I have a few minutes to kill before class or a meeting, I play it. My characters usually suck, though, because they have all the skills, so’s I can play with the fun graphics!
Ultima Online is very addictive. 7xGM Dex Monkey Anti-PK!
Tropico!
After four years I am still hopelessly addicted to Starcraft. I play it almost every day. Not on battle.net, but I play it alone…eight-way free-for-alls, one-on-twos…death! Destruction! Murder! Mayhem!
Guess what I’m going to do when I’m done reading the boards this morning? 
Europa Univeralis, mmm!
EU fans will be thrilled to hear that Paradox will be using many of the same concepts to create new games like Legion (EU in ancient Rome) Crusader Kings (EU in medeival times) and Hearts of Iron (a big WWII game)
Civ II
upgrading to II now - RingTFM
Egads! no one has mentioned QUAKE3?
Greatest game ever created.
IMHO of course
Jedi Knight 2
Paradox making a WWII game? Sweetest of sweets.
Anyway, right now I’m playing:
Morrowind. Can’t stop. The world is so real and deep that it’s like being in another reality altogether.
Oh, and those habit forming pushers at Firaxis announced today an expansion pack for Civ3- which will include scenarios, 8 new civs, and multiplayer.
I was addicted to Deus Ex. I’m currently addicted to Jedi Outcast. I will soon probably be addicted to Morrowind (which I’m not picking up until I get my new computer in two weeks. Yay me!)
Grand Prix Legends, almost every night lately. If you consider that a game.
Ahem…Baldur’s Gate 2, Dark Age of Camelot, Tribes 2, Diablo 2 expansion, and of course, Snood. 
Joemama: I remember Subspace! People still play that? Time to break out the “old” cd binder!
Civ III and Rollercoaster Tycoon. Both of which were bought for my kids. One game of each to check them out, and I was addicted.
Count me as another Civ III addict.
It’s not that I play every day, just that when I sit down to play for a couple minutes, I realize 8 hours later that I probably should have gone to my classes that day, or slept that night. I’m pretty sure if the building were burning down, I’d wait to flee until after I finished my bombing attacks on the German empire (I’m new at this, can’t get my technology past early airplanes). I mean, they only have 4 cities left. I’ll be done in 2 minutes, really.