It is difficult to limit it to 10, but I will try, some may be in my list solely for the online aspect, others for being deep single player games.
1. Deus Ex (the original for the PC)
2. Half-Life 1 (much love for the other games but the original practically shaped my early years as a gamer, I was 12 when it came out and it was one of the best games I have ever played)
3. Ultima 8
4. Ultima Online
5. TES3: Morrowind
6. Final Fantasy 10
7. Persona 4
8. Counter-Strike (I will go ahead and lump the GoldSRC versions, Source and GO all in one here, I have equal love for everything in the series)
9. Chrono Trigger (I practically lost myself in this game in the mid to late 90s)
10. System Shock 2
Ugh, that was very difficult, so many great games that I left out. I will note that I am also an enormous fan of the S.T.A.L.K.E.R series, Bioshock series, Batman Arkham series, Halo series, Resident Evil series, Doom games, Wolfenstein games, Duke Nukem 3D...sheesh.
I tried to list games in my top 10 that had a very special influence on me, though. While Wolfenstein 3D was the first ever shooter game I played, Deus Ex and Half-Life 1 were probably the most influential to me, games that really sucked me in and didn't release me. And then in the 90s, Ultima 8 was a game I played to obsession, while I later discovered The Elder Scrolls series, I didn't play Arena or Daggerfall in the 90s, I somehow found the Ultima games instead, which caused me to go on to play Ultima Online A LOT between the years of about 1998 to 2002, when a lot of my friends from that game started playing Dark Age of Camelot instead so I followed them to that game, then later to WoW. UO got me into MMORPG games though and made me feel like I was existing in another world.
Oh one honorable mention is the Half-Life mod Natural Selection, I think this originally came out in 2002, I remember playing it a lot in high school, really cool and unique game. A sequel came out in late 2012 on Steam.