I don't have any memories of the first game I played, but the first one I owned was Super Mario World on the SNES. I had that game so long that the battery died. Wouldn't keep my saves unless I played it (or at least put it in the console and turned it on) at least once every three days.
Now that I have it as a ROM, however, I can apply IPS patches and play all new versions of the game! ahhhh youth revisited.