This is a list of the video games which influenced me most. This means they were very good games and in addition contributed something to the gamer I became or to the person I became. The list of 5 best games is different and I'm not sure I could choose only 5 games.
I have been playing video games for a long time, since I was about 12 years old. I had a NES, and the cartridge with Super Mario Bros and Duck Hunt. But those games only captured my attention fo so long. I played a few other games too, but most were either too difficult or boring.
The first game which got me really hooked was The Legend of Zelda. I explored the whole map, heck, I _drew_ a map of the game, with secrets and everything. Twice, because I eventually lost the first one. I won't explain why Zelda was a huge game, you all know what game it is.
Skip ahead a few years. I eventually had a SNES, and Street Fighter II Turbo, and I loved that game. But I won't consider it my most influential fighting game, because honestly back then I was mostly a button masher.
No, that crown would go to Fighting Vipers, an excelent fighting game for the Sega Saturn, by the same team which gave us Virtua Fighter.
That game was, compared to most fighting games, slowly paced. And in that game, being a button masher would invariably get you killed. I haven't played any fighting games between Virtua Fighter II and Soul Calibur IV, but from my point of view all modern fighting games learned a lot from Fighting Vipers. It taught me to evolve beyond being a button masher, something which Virtua Fighter II couldn't do for me.
Skip ahead several months. I bought Panzer Dragoon Saga. That isn't just the best RPG in the Saturn (God rest her soul). No, it's probably the most awesome RPG until then. Take this with a grain of salt, though, since it's the second RPG I ever played. I loved it because of the universe, which captured my imagination for years to come. I was a very well read boy back then, so that's a huge thing to say. In hindsight, I would have sold an organ for that game, and it would have been worth it.
Honourable mention: A couple years later, I was hooked by Lords of Magic for pretty much the same reasons I was hooked by Panzer Dragoon Saga. It was so influential I even DM'ed a Dungeons and Dragons campaign on a universe I created inspired by it. It doesn"t get the title of Most Influential Game because it's too similar to other chosen games. I'm sincerely sadened by the fact that the game got a poor reception, but it was so full of bugs that I think it deserved it. The honourable mention goes to LOM: Special Edition, actually, because it no longer had bugs, it had better music, better visuals, and lots of added functionality. It encountered little success because the harm was already done.
At more or less the same time, I discovered Age of Empires. I really enjoyed the concept of the game, and this is the game that introduced me to the RTS genre. I also loved the following games and its expansions (I never played Rise of Rome, though). The game which really taught me the finesse of RTS games was Age of Empires III, because there are numbers everywhere. It's no longer about throwing teutonic knights and trebuchets at your foes until his base is no more. Now it's about balancing military and economy, choosing the right counters for what's in front of you, all that with smaller, more manageable armies. I'm by no means an awesome Age3 player, but I'm no slouch either, and that's thanks to the third in the series.
Finally, there's Baldur's Gate. Actually, I liked in Baldur's Gate a lot of what I already liked in The Legend of Zelda, but there are so many new things, and I love RPGs so much, that the RPG genre gets 2 wins in my short list (anyone who suggests that PDS and BG are the same genre will be beaten to death with a wooden club - JRPG and CRPG are very different). I also went on to really enjoy Neverwinter Nights, Mass Effect, Fallout 3 and all that, but Baldur's Gate is the game that started it all (Fallout and Diablo were released a little earlier too)
To sum up, the games which influenced me most are:
The Legend of Zelda, for a game with lots of exploration to do and time to invest
Panzer Dragoon Saga, most awesome JRPG in the goddamn Universe. Ever. Period.
Fighting Vipers, for teaching me fighting games.
Age of Empires III, for teaching me how to play RTS games well
Baldur's Gate series, because it's the earliest manifestation of the RPGs I most enjoy today.
Honorable mention to Lords of Magic Special Edition for the captivating setting and excellent game mechanics.