Hmm, there's a few that I love the most, but some of them aren't 'classics'
Shadow of the Colossus
I remember playing this for the first time during 2005, I think
This is, in my opinion, one of the greatest games ever made, and therefore my all time favorite and most important to my life in gaming. The story itself fits really nice with the actual theme of the game, which is defeating the 16 colossi to bring back your dead lover(Or close family member, since it never really tells you, which is left for speculation). Wander, the player, goes throughout the forbidden land in search of the bosses to slay, and each one is different and highly fun to take on for the first time. I remember when my father introduced me to it on the PS2, and I was blown away by the graphics(at the time) and the sense of scale. You are but a man and his horse fighting against impossible odds.
For being on the PlayStation 2, the actual world you played in was massive! I mean, you could simply go anywhere in it, although you had to fight the colossus in order Dormin, who tells you what you need to know and is a God/Demi-God, shows you.
I'll admit, it was the first and only game to actually make me cry at the end. It was that good. I'll never place any game above this title, no matter what anyone tells me.
Uncharted Series
I first played Uncharted during the summer of 2007, if I recall correctly.
It was basically my first platformer, save for the originals of course. I absolutely loved the mix of climbing, action and my man crush, Nathan Drake! It was really the first game to get me into shooters and third-person games, which I got sick from, funnily enough.
The second of the series, Among Thieves, is still my favorite of the genre. Sorry Lara Croft, but I never really got into your games. It paved the way to some other good games, as well as actually getting me fully into games, which at the time I only played them on occasion
Resident Evil 2
Played this in 2006 on the Wii
This game is my favorite Resident Evil, as well as my top pick for horror games. You just can't beat the T-Virus, ladies and gentlemen.
I don't remember much about it, but I do know I was completely out of it as I played it. Leon, Clair and company were legends to me, and I don't know if I could look at any RE game the same way again. I'm looking at you RE6.
I remember calling Mr.X "Trenchy", which was an inside joke with my Mom and Dad, who were gamers as well at the time. The memories of fighting the man in the trench-coat were good indeed, and I'll never forget it.
That's mainly it, and there are plenty more, but I figured a top three would do nicely.
:D