I've played hundreds of computer games over the years starting with simple pong on the TV.

Most of the games I have played have been for the PC, with only a few (quite recently) being for the Playstation 2 (and even then, only because my kids got one for Christmas).

Since I like making lists so much, here are some lists of games I've enjoyed - and played right to the end. 
Oh, and at the end of this page, I'm providing links to some great FREE games that you might want to download (for the PC)

PC Games

Adventure Games
Role Playing Games
Platform Games
First Person Shooter Games
Other Miscellaneous Games
Things I haven't finished

Playstation 2 Games

All finished so far
Things I haven't finished

Free Games

I'll have to think of some sub-categories before I start putting things up en-masse however here are a few good games to get you started.

Adventure Games

• ADG Interactive: Remakers of King's Quest I and II and Quest for Glory II: Trial by Fire
• Infamous Adventures: Remake of King's Quest III
• The Ultimate SCI and AGI Site (Remakes and Fan Games)
  

Platform Games

• Open Source Remake of Commander Keen
  

Shooter Games

• Doomsday: Open Source Port of Doom, Heretic and Hexen Engines
  

Role Playing Games

• Ultima Remake Site
  

Online Games

• Lots of Quite Good Online Doctor Who Games
  
• Remake of The HitchHiker's Guide to the Galaxy Text Adventure Game

Assorted Sites

• Video Game Remakes Site