Tough question. I would say the PS2 era when I was a young teenager, mostly because of how less I had back then compared to now, where I have this large library of games on an internet account that I never feel thankful for.
Back in my PS2 days, I appreciated all the small things that I had and always dreamed of having more. I played Mortal Kombat Deadly Alliance 24/7, unlocking everything in the krypt and memorizing every character's combos and fatalities. I drowned myself in the game's mythos, going in my backyard and enacting the character's fighting styles; I asked my mother to buy me a bamboo stick at my nearest Home Depot so I could use it to swing at the air, where every inch of movement made me feel like a kung-fu master.
I Mortal Kombat Deadly Alliance. That game was fucking magical.
Then I encountered the Final Fantasy series. I first played ten, thus beginning my now missing cherishment of Japanese role-playing games. In fact, I could say that Final Fantasy X was the first role-playing game that I've ever played, making it personally my most memorable thing ever. A friend and I would just sit down and experience FFX, despite one sitting down and listening and the other playing. We would congress about what one should do during a turn-based combat event and discuss how much we wanted to fuck Yuna (we were closeted weirdos, okay?). I will never forget it.