Super Mario Bros 3 is probably the first game that comes to mind as I was finally old enough to really get excited for specific games, and that one was hugely anticipated at the time. The whole game blew my 7 year old mind (think thats the math at least). Up until that point it was mostly grabbing games based on their box art at the rental place, or based on what looked cool in those displays in Toys R Us. And you played those whether they were good or not. That wasn't the case with SMB3 though.

Super Mario World is probably the next example, for similar reason as above but even bigger and better. Then into the Donkey Kong Country games. Final Fantasy VI was another game that blew me away when I first saw it. That intro was crazy when you still considered the SNES a new platform. Pokemon Blue for the Pocket I got specifically to play it.

Played lots of the original Diablo, though I was terrible at it back then. Actually didn't like Diablo 2 that much when it was first released, but got into it again years later. Baldur's Gate 2 was fun for a while, but never beat it (and didn't play the first until much later). Lords of Magic was a game I played the crap out of but was terrible at the game as well.

If we are just talking about time played then games like Baseball Stars, Excitebike, Track and Field with the Power Pad, etc. You just played the shit out of every game you owned back then. Because you only got them once or twice a year.