What comes to mind are the following (no coherent order, sorry):
Adventure games: Maniac Mansion (C64/NES), Zak McKracken & The Alien Mindbenders (Amiga), The Secret Of Money Island (Amiga), King´s Quest series (Amiga), Space Quest 2 (Amiga), Deja Vu (Amiga), It Came From The Desert (Amiga), King Of Chicago (Amiga).
Platform games: Buble Bobble (C64/NES), Mario series (NES), Megaman 2 (NES), Wonderboy In Monsterland (Amiga), New Zealand Story (C64/Amiga), Duck Tales (both NES & Amiga versions, which were completely different), Frogger 1 & 2 (C64), The Great Giana Sisters (C64), Moon Patrol (C64).
Action games: Shinobi (C64), Contra (C64), Gryzor (C64), Ikari Warriors (Amiga), Cabal (Amiga), Gauntlet 1 & 2 (C64), Hysteria (C64), Aaargh! (C64), Rampage (Amiga), Golden Axe (Amiga), Aaargh (C64), Rampage (Amiga), Space Harrier (C64)
Fighting games: Double Dragon (C64), Street Fighter II (Amiga)
Shoot 'em ups: 1943 (C64), Silkworm (Amiga)
Strategy: Defender Of The Crown (C64/Amiga), Pirates (Amiga), Civilization (Amiga)
Simulation: Wings (Amiga), Destroyer (C64), Silent Service (Amiga)
Racing: Pit Stop 2 (C64), Lotus Challenge 1 & 2 (Amiga), Buggy Boy (Amiga), Turbo Outrun (Amiga)
Puzzle/Skill: Boulder Dash series (C64), Krakout (C64), Pang (C64/Amiga)
Other: Law Of The West (C64), Impossible Mission 1 & 2, (C64) H.E.R.O. (C64), Paperboy (Amiga), Typhoon Thompson In Search Of The Sea Child (Amiga), Spy vs. Spy 1 (C64) & 3 (NES)
Wow, it really brought back a lot of memories trying to remember the games of the past. What struck me is how much time I wasted on games that I didn´t actually enjoy. Some games, like The Last Ninja, I thought were cool and so I wanted to like them but they were mostly irritating since I was bad at them and didn´t understand how they worked (you know, effective fighting strategy, enemies' weak points, which order to do stuff). The manuals back then were sparse at best and no internet to get tips from (I had few friends and they were even worse than me at games so they couldn´t help).