I'm just listing games that I'd easily pick up and play right now. Games that are at the highest quality of playability, addictiveness, etc. My numbering is there, as well as reasoning. I'd recommend getting these games if you haven't already.
Hunter (Totally 3D, total freedom of movement, 10+ vehicles, dress as the enemy, a 80 km2 world, ride helicopters, swim, hunt animals. And made in 1991.
Carrier Command (A "Bush 'em Up"
. Destroy enemies, drill for oil, all over the world. Addictive.)
High Steel (Original game, construct buildings in a puzzle kind of way.)
Sonic the Hedgehog
Sonic the Hedgehog II (probably the first awesome co-op mode ever made.)
Streets of Rage II (4) (see above, still totally playable even to this day.)
Adventures of Alundra (7) (Zelda: ALttP clone to some, but never-the-less an awesome "five-hours-passed-in-thirty-minutes" experience.)
Final Fantasy VII (1) (It affected me on an emotional level. My first RPG, but it was such an advanced game for the mid 90s.)
Final Fantasy VIII (6)
Metal Gear Solid (2)
Resident Evil (5)
Resident Evil II (3)
Super Puzzle Fighter II Turbo (The first time i played this it was for 9 hours. I thought it gave me epilepsy. Addictive and extremely tactical.)
Legend of Zelda: Ocarina of Time
Devil May Cry (Most fluid combat system I've seen.)
Metal Gear Solid II: Sons of Liberty
Silent Hill 2 (Multi-layered, multi-textured and multi-ending storyline. Plus it's sexually aware without being childish.)
Eternal Darkness: Sanity's Requiem
Resident Evil 0
Soul Calibur 2
Alien versus Predator II
Call of Duty (9) (One of the few games to make every freaking hair on my body stand up. The Russian levels: play them now.)
Deus Ex (8)
Max Payne II: Fall of Max Payne (10)
Soldier of Fortune II: Double Helix (One of the best stress relievers you'll find.)
Star Wars: Knights of the Old Republic
Any gaming site you go to, any person you talk to, and most people will all have different opinions on "the best games." We're more united on the worst games. I wonder the opinions of people here on games such as Daikatana and Mortyr...
I've not listed the overly-obvious. I work in an office, I'm tired of typing.
For the Atari ST games, i grew up on that stuff. If you are interested in games such as Hunter, i suggest you "hunt" them down as they are awesome and over-looked precursors and pioneers of games such as Grand Theft Auto.