Just to name a few notable examples:
The Legend of Zelda: The Wind Waker - Credits Theme [http://www.youtube.com/watch?v=eoWlHcqznRA]
This might seem like a bit of an odd choice, but this song easily represents what games should be: a fun adventure. Wind Waker doesn't get a lot of the respect it deserves because of it's graphical style, but the spirit which makes the Legend of Zelda series the most renowned in gaming is still there. Too many people are too quick to try and grow up (often leaping at more "mature" games), but LoZ:WW aims to please the inner-child is us all.
Chrono Trigger - To Good Friends [http://www.youtube.com/watch?v=S5F3nbbdlv4]
The title of the game very much speaks for itself, it's a tune which says farewell to all the characters seen in CT. The use of the musicbox at the beginning is innately nostalgic, and it's nothing short of fitting. Chrono Trigger is a fun adventure, but like all things it must come to an end; "To Good Friends" effectively it's own farewell to the player.
Final Fantasy VI - Aria de Mezzo Carattere [http://www.youtube.com/watch?v=KuvSDedrtKM]
The legendary opera sequence from FFVI is easily one of the greatest triumphs for video game music, it's too beautiful to describe. The Final Fantasy series is often regarded as having the best music of all of gaming, and among them FFVI is considered the best of them all (many of the other pieces from the game are certainly worth of being in this topic, and I'll give a quick nod to Forever Rachel [http://www.youtube.com/watch?v=-TzPDsBNtsw] in particular). However, if there's one tune to pick out of everything in FFVI... it's the opera, hands down. While the previous two songs I posted remind us that what games are really about, the Aria de Mezzo Carattere reaches beyond that to show us that games can be more than that as well.