There are so many ones to choose from! Perhaps this topic will introduce everyone to some great titles they've never played before. Here are mine:
1. Skies of Arcadia/Legends Dreamcast and Gamecube: No game has ever captured the spirit of exploration and discovery like this game, (and it's from Rieko Kodama, one of the PS creators.) There are so many facets to this game that I never manage to tire of it. An amazing soundtrack, (especially on the DC) monster battles, ship battles, crew, discoveries, many side quests, and most importantly, the positivity of the philosophy of the main characters of a goal-driven life in the pursuit of happiness makes this one my all-time favorite.
2. PSIV: Not much to say here; you all know why I love it! Replay value, character development, and the climax of the series!
3. Star Control 2: the best space adventure game out there! It's now Freeware thanks to the Urquan Masters Project with more options, internet melee play, (though Super Melee never gets old even without the internet,) and the single player game is still timeless, difficult, loads of fun, and utterly hilarious!
4. PSII: The game that introduced me to the series. I was riveted from beginning to end, and the party selection has given me endless replay value. I can still think of a dozen new ways to play through it!
5. Madden 2004, XBox: the last great football game before EA cannibalized the rating system to compensate for online play. The Franchise mode was great as it rewarded a player with long-term vision.
There are so many other great ones that barely missed the cut, (Starcraft, Fire Emblem 7,) but I've got to stop someplace or I'll be rattling on all night!