The first computer I had was a Vic 20, and the game I remember was a very primitive racing game, where the road was defined by just white posts along the sides. Can't remember what it was called at all. It was on cartridge rather than tape - we did have some games on tape, but get the Vic 20 second hand with no instructions, and none of us had the faintest idea how to load them!
I had Bruce Lee on the C64 a few years later, and it was a great game. Having a friend take over one of the enemies (the Sumo guy as I remember) to help you out, but they would invariably get bored and start attacking you.
And when you finally completed it ... it just started from the beginning again.
Does anyone remember from the C64, the first game I ever came across with licensed music (assuming it wasn't just stolen music) a side scrolling r-type-type which had an instrumental of I Feel Love by Donna Summer as the background music. Because it was very electronic disco music anyway, it was quite a faithful rendition.
Other games I remember really fondly are Kickstart 2, Racing Destruction Set, Wizball. I'm sure there are others - I had hundreds (all bought for £1.99, mostly from WHSmith - those were the days!)