Games that had meaning, other than for their gameplay.
Which games have you got an attachment to and that tell a story, other than for just being 'that game'? Here are mine:
Wonderboy - I first remember seeing Wonderboy when I went to a school trip to Valkenburg in Holland. I was sitting with my friends in the hotel and looked over to see the girl I really fancied playing it on an arcade machine. (OK, not really amiga related).
Settlers - The sheer amount of time I used to spend on this game with my best friend. When we were early teens my friend would stay over so we could just play Settlers. We lost track of time most times and it went on till about 2am in the morning, split-screen, me and him with the mouse, eyes fixed on the monitor! Not only that, we made comedy disks for the Amiga - Fun times.
Wolfchild - This is what inspired me to get an Amiga. My uncle had an Amiga and I always wanted to play this around his house, although I think the game isnt all that good now. I saved up my pocket money of £10 a week for ages, to buy myself an Amiga 500.