I would also add the following:
Deus Ex: My favorite game of all time. The stuff you can get away with in this game has only been matched by, as far as I know, Fallout 3/New Vegas. There's tons of optional dialogue that you don't have to listen to (but fleshes out the world so well), you can do all manner of violence to enemies and NPC's (potentially playing it as a sadist if you want to), the game gives you alternate lines/conversations if you do things the game doesn't expect you to (like kill a major character without any prompting from the game itself), tons of different ways to solve missions, non-lethal options, great character development...the first time I played it, I was blown away by the fact that I could redo a mission, try something different and be rewarded with different dialogue/rewards for my ingenuity.
Pokemon Gold/Silver: To add to the above, finding out that you could revisit the Kanto region, see how it changed and win all of the original badges again (from stronger trainers) was and still is amazing. It was like a whole other world opening up to the player when they beat the game, albeit a revamped version of the original.
Final Fantasy VI: I could probably just rattle them off. Shadow and everything about him (waiting for him, the dreams, Relm's affiliation with him), the opera scene, Gogo, suplexing a train, etc. Yet, the moment where you get the second airship back and come up through the ocean as "Searching for Friends", and having full rein to go around the world map and see the devastation, is just one of those magic moments that is impossible to replicate after the first time.