Hmmm my memory of my childhood is a little spotty so some of these may end up being when i was a teenager.

had a ps1 then later a PS2
for the PS1 it was absolutely the PS1 Spider-Man games (Spider-Man and Spider-Man 2: Enter Electro) , little charming things fun little stories and levels.

for the ps2 it was Yu-Gi-Oh! Duelists Of The Roses one of the earlier games that elected to use the cards as a basis for playing a different style of game over being a Simulator for the Main card game.

(must also shout out Dark Cloud while i wouldn't actually own and finish it till my late teens, it was on one of the Demo Disks i had and i played that demo a fair bit)