Here are some of mine:
playing through Secret of Mana together with my brother. We even got up at around 7 am in the weekends just to get a couple hours of SoM in before breakfast.
More recently, I've been blown away by Persona 3 (currently playing part 4) and Kingdom Hearts (part 2 was rather a dissapointment for me). These are both games I really connected with, and which sucked me in; as such, I've strong memories of them.