ps2: Metal Gear Solid 2 and 3
Gran Turismo 3 and 4
Kengo:Master of Bushido
(most of you may not have heard of this one and probably would not like... . game play wasn't very good but loved it non the less for personal reasons)
PS1: MGS1 (although I never owned a PS1, but I did buy MGS1 on the PS Store).
PS2: MGS3 (Amazing game, it made me shed manly tears... MGS2, GoW1, and GoW3 are up there too though love those games).
Ironically all my favorite PS1, and PS2 games I bought and played on PS3, when I actually played PS2 I didn't have very good judgment when it came to games.
My favorite PS2 game that I played when I actually had a PS2 would be Monster Hunter and Driver Parallel Lines (Closest I'd ever get to a GTA game because my parents didn't want me playing GTA XD... I was in middle school or elementary school back then).
PlayStation: Final Fantasy Series (runner up, legend of mana, and yes i still have all the original ps1 dics)
PlayStation 2: FFXI most hours ever put into a game, ever.
MGS Collection (which also came with MGS1 ps1 disc, so its a two-fer!) runner-up, FFX FFXII
If I had to pick just one game for PS1 & PS2, then it would probably be either Vagrant Story or Castlevania: Symphony of the Night for PS1 and maybe Zone of the Enders: 2nd Runner, SSX3 or DMC3 for PS2.
Resident Evil 1, 2, 3: Nemesis
Oddworld: Abe's Oddysee
Silent Hill 2, 3, 4: The Room
Project Zero 1, 2: Crimson Butterfly, 3: The Tormented
Devil May Cry 1, 3: Dante's Awakening
Shadow Of The Colossus
God Of War 1, 2
Rule Of Rose