To clarify what I'm trying to say is, what games in the beginning have set things up in a way where you instantly think: "Wow, this is some really exciting content and a great introduction to game and what's to come." Kind of letting you in on the scope of the particular world and story just from the initial couple of hours being in-game.
Final Fantasy VII (Only played this one recently & introduction/set-up in this game was great. The beginning content in Midgar was very well put together and exciting. Really gave you a good grasp of what was to come.)
Kingdom Hearts II (A lot of people find the Roxas opening stuff to be very boring, but I think it is still to this day one of my most favourite introductions to any video game. The lead up to playing as Sora is what really makes it too.)
Final Fantasy X (I liked how straight off the bat you got an idea of Zanarkand & Tidus's backstory. The introduction of Wakka and the others + intitial gameplay on Besaid was fun.)
The Legend of Zelda: A Link to the Past/The Legend of Zelda: Windwaker (Both great introductions but I always loved how in ALTTP you'd wander up to the castle while it's storming. Sneaking through the secret hole in the ground and picking up the sword and shield from your uncle. Windwakers set-up was good. Running around town and throwing pots at people was always a laugh too.
Other mentions: Rayman 3 Hoodlum Havoc, Mario 64, Mario Odyssey.