Earthbound - Beautiful music, not all orchestral or hard rock like most other games, based a lot around jazz, swing, and early rock/blues, it definitely caught my ear.
Pokemon Black and White - A more recent example, but it's music was really, really good, AND if you talk to certain people in towns, you get extra instruments to join in the music, which just blows my fuckin mind.
Mother 3 - But for different reasons than Earthbound. In this game, battle mechanics influence how well your attacks do. Each character has an instrument that they are represented by. When you use that characters attack, if you tap the 'A' button in rhythm with that instrument, you get bonus mini-attacks. Again, blew my fuckin mind.
Final Fantasy 6 - This game has great compositions and excellent ambiance. But that's not really why I liked it. It taught me the effectiveness of a leitmotif, or a recurring melody/musical cue.
Mega Man 2 - Oh man Mega Man 2 rocked my socks.