One of the first games I remember getting was Pokemon Yellow, and that whole game made a huge impact on me, but the music especially. There were many times where I'd just do nothing but just let the music go - and that's why I stood still on Route 4 for hours, or fell asleep to the sounds of Saffron City (yes, the battery died... it sucked at the time but hey as a kid you have plenty of time to make up lost process).
Other music that caught my attention as a kid: Crash Bandicoot, Jet Moto (hell yes whoever mentioned it), Super Smash Bros., Donkey Kong 64, Banjo Tooie, and Majora's Mask (I didn't play Kazooie or Ocarina until later in life, go figure). Oh and the Metal Gear Solid OST... I never played the game itself until my late teens, but that soundtrack burrowed its way into my mind just from hearing my step-dad play the game. Even now, that little Konami jingle (the Policenauts one) gives me nostalgic shivers