Duke Nukem 3D, since that was practically the only game I had on my pc for years, along with a demo of Rise of the Triads, after that Age of Empires I & II.
As I got older, I got addicted to Hitman, and now I'm embracing Death to Spies and S.T.A.L.K.E.R.
For consoles, I remember I was really addicted to Sonic the Hedgehog and Jurassic Park on Genesis, along with Cannon Fodder and Golden Axe II.
Tenchu, Metal Gear Solid and GTA2 were my addiction on PSX, when the PS2 arrived I got really addicted to Romances of the Three Kingdoms IX I think... or was it VIII?
I guess the only current generation games I was ever addicted to were Red Dead Redemption (Big spaghetti western fan here), Arkham Asylum and Mass Effect.