Pokemon (1st gen and 2nd gen -- plus some of the spin-off stuff from that era to a lesser extent) captivated me in a way no other games have before or since. Honestly, I hope nothing ever grabs me that hard again; it's kind of scary how obsessed I was in retrospect.
A number of N64 games also made a big impact on me. Majora's Mask was probably my second favorite game (after Pokemon) for many years. (OoT made an impression too, but I played MM first.) I loved the music in Diddy Kong Racing. I got 120 stars in Mario 64, and when I tried it again as an adult, I really appreciated how short and to the point levels could be (not that I played that way as a kid) -- also the camera in that game sucked. Castlevania: Legacy of Darkness kind of disturbed me a bit as a kid, but it's probably the first game I encountered a sort of "New Game Plus" in, which was neat. (People have since told me that's the "black sheep" of the series and that it's really weird that that's the only one I've played significantly.)
Duke Nukem 3D was the first game I modded, I think (very simple graphical stuff). Definitely wasn't age appropriate but I played the heck of it anyway. Didn't really get much into other shooters other than playing through the main game of Perfect Dark on N64 and playing split-screen Golden Eye with friends.
I also played a lot of Sim<Whatever> games -- particularly SimCity 2000, SimEarth, and SimTower. Also had a bunch of others like SimFarm and even some of the more obscure ones like SimSafari. Streets of SimCity and SimCopter being able to load SC2K maps was really neat though. Played a fair amount of other city builders and simulation games like Caesar III and Roller Coaster Tycoon too. My parents probably hoped I'd become some sort of business manager. :p
I had a lot of creative tools back then as well which I treated as not-that-different from video games. Various Kid Pix programs (one of which had a bunch of odd video clips integrated -- including a short documentary about jackalopes of all things), Kid's Studio, Digital Chisel, some version of HyperCard, etc. Game Maker -- which I found around the year 2000 back when it was still on www.cs.uu.nl -- ultimately led me to being a professional programmer.