For Playstation 1, my best memories would have to be playing the Crash Bandicoot games. Really wish there were proper Crash games on the current consoles (I have Crash of the Titans, and it was fun, but it still wasn't nearly as good as the original 3 games on the PS1).

As far as the Playstation 2 goes, there are a lot of games to pick between. The first ones that come to mind are Grand Theft Auto III, Vice City, and San Andreas (even though I never finished the story on any of the three). No other games gave the player that kind of freedom at that time. Another couple of games that I really enjoyed playing during the PS2 era were Tony Hawk's Underground 1 & 2 (I played one of these games almost daily for years), and Downhill Domination (which is a game I don't feel gets enough attention, I wore a demo disk of Downhill Domination out before I found a copy of it used to finally purchase).

My best memory was in the time of Ps2,
I used to play Ape Escape 2 with my sister and we used to have loads of fun.
I still like the Ape Escape series, its a shame they don't make it for Ps3 or Ps4
I think lots of people would've liked it.

My best memory of the playstation was playing Grand Trismo for the first time. And I am not talking about playing Grand Trismo 5 or 6 or any of those, I am talking about play the first one.