First I owned would have been those AA battery powered handheld games that existed probably before you were born... I remember in particular a Teenage Mutant Turtles one and a watch on which you could play a Mario game. This was the only Mario game I would have played as a child cos I never had a Nintendo, although my best friend at the time did.
First proper game I owned was Sonic the Hedgehog 2 when I got a Megadrive for Christmas. Dunno if the game holds up now but I absolutely loved the music in it and the graphics, it's all massively nostalgic for me now. Had various other games on the Megadrive, including Mickey and Donald's World of Illusion (which still somehow haunts me), FIFA, Sensible Soccer, Brian Lara's Cricket, James Pond: Robocod, Toejam and Earl 2: Panic on Funkotron (great soundtrack), Predator 2 (pretty sure this was a terrible game but fed my appetite for grownup violence)... My king of games on Megadrive was Street Fighter 2, which I was forced to save up for by my parents (at the time a Herculean feat).
Next either came the PC or the Playstation, I can't recall which now. On PC I played Unreal Tournament a lot, also Rainbow Six, FIFA World Cup 98... I played the Half Life demo a lot cos I didn't own the game. My friend at school had a PC and I played Half Life proper on that, as well as the original GTA/GTA2, etc. Counterstrike. Loads of games. This is actually giving me nostalgic melancholic feelings for my teenage years. My favourite PC game is probably Grim Fandango, which I used a walkthrough guide to cheat my way through.
I owned a PS1, can't remember the first game I had on it. But big ones included Tekken 2 and 3, Resident Evil, Final Fantasy 7, Ace Combat 2... FF7 was absolutely amazing (up until I got stuck) but the king of PS1 games for me was Tony Hawks Pro Skater, which I began playing as I began skateboarding - and the sequel, which was probably the best of all. The soundtrack is soldered into my brain. I particularly enjoyed going round and round and round the levels, doing tricks, while listening to CDs and Tim Westwood's Radio 1 Rap Show.
Sighhhh...
The history doesn't end there but I'm prang I'm gonna delete this post