Here are my 5 most influential games in no particular order. As you can see most of them are pretty old games I played early in my gaming career. I chose them since I think these are the games that made me the gamer I am today.
The original start of the greatest series of action RPG's in existence. One of the first games I bought for the first PC I ever owned. A cyrix 5x86 dx 100 budget pc from the local hardwareshop. Paid for by with money from my newspaper round. I literally spend hundreds of hours on this game, most of it alone since I didn't have internet access back then. By the time I did get access to the internet I learned about "variant-play". Which means voluntarily taking extra restrictions to the build of your character in order to increase the difficulty of the game. The best known variant, and also the first one I played, was the Beyond Naked Mage (BNM). The BNM refused to wear any armor unless it was cursed. Cursed meant the armor had stats which were harmful to your character, like negative stats or resistances. This made me a better player adn drove me forward in other games to look for opportunities to make the game more challenging in ways the creators had not foreseen or intended.
Contrary to what the game selection says I did not play this game on PC. I played it on a friends Amiga 500, but that option is not available in the game list. This game got me into playing RTS games for the first time. It was single player only and had three factions to choose from, the houses of Atreides, Harkonnen and Ordos. I played together with Leonard who owned both the game and the Amiga workstation and I have fond memories of dreaming up strategies and arguing on the way they should be executed.
The original edition sold on 6 3.5" floppy disks. I had played Wolfenstein before this, but i didn't like that as that game got me headaches after play sessions longer than ten minutes. The cool premise of being a lone spacemarine in a spacestation overtaken by whole armies of monsters has entertained me for hundreds of hours. I think I have completed this game at least 10 times including 1 run on nightmare difficulty. Of course I also played the remake Doom 3, but that didn't do much for me for some reason.
Sid Meier's Civilization III:
The game that got me into empire building / turn based strategy games. Spending over 40 hours on a single map trying to get my own civilization to world dominance and then start again with another one on a randomly generated map. I even spent time thinking up strategies to take out the next empire while I was away from my computer. Loads of fun and a guarantee for endless entertainment. The lates version, Civilization IV, is even better and I still play that game every now and then.
The Legend of Zelda: Link's Awakening:
My first exposure to the magic of Shigeru Miyamoto. A wondrous and surprisingly large world contained on a single Game Boy cartridge. I've lost hours in this game exploring the map, solving puzzles and talking to all the npc's. The best part was that every time you concluded a dungeon you got a new piece of gear, mostly weapons. More often then not you could use these to access area's that weren't accessible before. This lead to a near endless amount of backtracking and exploring all the area's in the game. Sadly I lost the cartridge and I'm still regularly found at several retro gameshops to try and find a working copy so I can revisit it.
Role-playing/Isometric Action RPG
Role-playing/Third-Person 2D Action RPG
Strategy/2D Real-Time Strategy