That moment in the first Unreal when you go from the ship you start off in to the outdoors...it may have lost a lot over the last twelve years, but that was a moment where you could feel awe at the world you were playing in.
The first Mass Effect game, several times, the first of which was probably less than an hour in.
Disgaea, when you start to realize how much there is to do beyond the main storyline.
Final Fantasy 7, if you can get past the translation and blocky graphics, as you being to understand the scope of the plot and the nature of the antagonist. Final Fantasy 10, also, for that matter.
Xenosaga, if you stayed awake through that one hour-long cutscene. You know which one I'm talking about.
Halo, if you were better at it than I was and didn't get stuck in a corner looking up while the enemies ate you, cursing at the gamepad for not being anything like keyboard+mouse.
Star Control 2, or Starflight 2. They're kind of the same game, actually, and both awesome.
Quest for Glory 1: So You Want To Be A Hero. The original EGA version, back when it was called Heroes Quest, with text parser. Or, if you were more of a Kingdom of Daventry fan, King's Quest.
Fallout, or Fallout 2, or Fallout: New Vegas. Not as much Fallout 3 or Fallout: Brotherhood of Steel. Planescape: Torment. Baldur's Gate 2 + Throne of Bhaal. Big games with a big world (sometimes worlds) to explore and change, and the feeling that what you do makes a big difference, even if you don't start out as king badass.