I like this question! Hmm... let me think...
1.) The Legend of Zelda: Ocarina of Time: I was about seven or eight when I got this game, and by then, I'd been playing games for a long time. Having to use DOS to play
Hugo's House of Horrors (props to any and all who know what those two are because they remember them, not because they looked them up later on.

) is what taught me to read and write. But the game that
really got me hooked on gaming was
OoT. It's the first game I can remember playing every day, day after day, for a couple of years. It's the game that taught me to search the corners for secrets (remember picking up rocks to reveal holes?), to solve puzzles, to be patient when I got stuck, (I can't tell you how long it took me to figure out where the Kokiri Sword was, or how to get Lord Jabu-Jabu to open his mouth), and, most importantly, it's the first game I can remember going back to complete more than once (I beat it quite literally close to thirty times. Twenty-seven or twenty-eight, as I recall).
Ocarina of Time is the game that made me decide that I was a gamer, before I ever knew that you could classify yourself as such. It's still my favourite game of all time, still occupies the same special place in my heart that it did when it was new, and still fills me with unbridled nostalgia when I see gameplay footage from the original version. (I was close to buying a 3DS just for the remake, but that would honestly have been the only game I ever purchased for it).
2.) Super Mario 64: The Nintendo 64 wasn't my first console, but it's where all of my favourite gaming memories lie.
Super Mario 64 is the first game I ever completed 100% when it wasn't completely necessary, (and, let me tell you, opening that cannon outside Peach's castle and shooting onto the roof to see Yoshi is still one of my favourite childhood memories. It's just a shame that he wasn't playable or anything), and still stands as one of the few that I've actually 100%'d. This was also the first Mario game that I completed, so it's the first time I ever put that bastard Bowser in his place. It's also one of the first games that I can remember playing because I
had to beat it, not because it was fun, (around the time I got to the room with the Tick-Tock Clock stage, I suddenly didn't care about seeing the levels anymore: I just wanted to
destroy Bowser. I don't know what caused the shift, but it was alarming). This is also the last game I can remember playing on my Nintendo 64 before it broke. (Not the last one I purchased, but actually the last cartridge I put in the machine).
3.) Perfect Dark: This is the game that introduced me to first-person shooters, a sliding difficulty scale (I had always played games on the easiest setting before this, but the extra missions you could unlock based on the difficulty you played the rest of the game at made me change my mind for this game). This is also the game that made me thing that, just maybe, I didn't like first-person shooters all that much (some of the missions frustrated me to no end, especially on higher difficulties). This is the game that really introduced me to multiplayer: a couple of my friends and I played this game
to death, trying to complete all of the challenges, unlock every map, best each others records, and so on. My one friend and I still go back to this game occasionally on his N64, but it's nowhere near as fun as it used to be. She hasn't aged well, Joanna Dark.
4.) Paper Mario: Paper Mario was my introduction to turn-based RPGs, (I would say RPGs in general, but I've never been sure if
OoT qualifies, so I say turn-based, just to be safe). Turn-based RPGs are still one of my favourite kinds of games, so I guess you can say that
Paper Mario introduced me to one of my favourite genres. It's also the first turn-based RPG I played to completion, the first Mario RPG I ever played, and the game that introduced me to the concept of 'parties' in RPGs (which stands to reason, as it was my first turn-based one). It was also the game that started me wondering just
why Peach could be so ditzy: exactly how many times have you been kidnapped, Peach? Uh-huh, and you don't think that maybe beefing up security, or, you know, sending a tactical strike team to get rid of Bowser once and for all
might be viable ideas?
5.) Final Fantasy X: As a hardcore RPG fan, my first
Final Fantasy game was always going to be a big moment. Because it was my first,
FFX is also my favourite. I've gone back to play each game, but none of them can compare to
X, in my eyes. This is the game that made me realize that RPG bosses had to be considered strategically at times, introduced me to many of the cliches that plague RPGs, and is the first game I can remember that I played quite literally all day. (when I first got it, I played it from the time I woke up 'til the time I went to bed for nearly two weeks straight, with a brief break for food around dinner time). It was also the first game I got for my PlayStation 2, which made it an even bigger deal for me. It's the first RPG that convinced me that sidequests were worth the time, (I spent many an hour searching for how to unlock the extra aeons, and getting each Celestial weapon). Next to
OoT,
Final Fantasy X is my favourite game. When I learned there was supposed to be an HD remake, I was tremendously excited, but now I rather fear that that has fallen by the wayside. Now, I'm just waiting for a PSN release. Come on, Sony, if you can get
Persona 3 up there, you can convince Square to let you put
FFX on it.
In case you couldn't guess from that, the Nintendo 64 was absolutely vital to my becoming a gamer. It's also the last Nintendo system I owned. I really tried to come up with some games that
weren't on the N64, but it was a lot harder than I figured it would be. In the end, I guess it just goes to show how much the system meant to me. If I could pick a point in my life to go back to, it would be to when the N64 was still in its prime. Those were the days...