For me its not the Games themselves that made them my favorite, (As you will see by my list) its the moments in time, and the fun with friends that i had with them...
(Not in any order, Moreso for each System)
Sonic(Sega)
Crazy Taxi - Sega Dreamcast
NBA - Dreamcast
(Also have to say that its sad that the Dreamcast system went the way of the Dinosaur Because they were TOO ADVANCED for there timeperiod some of the ideas that they used didnt become popular till 10 years after Sega went under)
Mario 1 And 2 (NES)
Pong
Galaga
(First games i ever played on Atari 2200)
Grand Turismo 3-4 (Playstation)
Smackdown Vrs Raw Series (PS2)
Ninja Gaden (As well as a few others for Original Xbox)
Battlefield Series (360)
I Know these are all over the place, but were the ones that (For me) Defined Gaming, Becaue i could never narrow it down to just one.