Okami the first time I played it. I fell in love with the world and its characters so much, and the scene toward the end of the last boss fight was just so incredibly touching. I actually even cried a little because I knew that if we ever got more Okami it just wouldn't be the same because Clover Studio's shutting down had been confirmed by then. No other game had touched me quite so much though a few runners up would be Ocarina of Time and Chrono Trigger.
To this day I get choked up listening to the instrumental version of "Reset ~ Thank You," that scene was just so heart-wrenchingly uplifting. Still resides in my top pick for most emotional moment in a video game.
Anyway, this phenomenon affects me all the time and that has a lot to do with my own personality. Its particularly bad with short series like trilogies. Hell, right now I really want to play Danganronpa 2, but I know I'm going to be really sad after finishing it until I can play Danganronpa 3, or at least until I can afford UDG.
There was actually a tweet going around the other day that I think pertains quite nicely to this thread, it read:
"Have you ever liked a book so much you regretted reading it because you know you'll never again get to read it for the first time?"
Honestly, the game that had the worst such effect on me was Persona 4. They did this amazing job bringing Inaba to life and really drawing you into Yu's year there, and then they threw that simple goodbye ending at you and it really felt like a personal chapter of my life had come to an end. It's why I feel so strongly against the addition of the epilogue in P4 Golden, it really does kind of ruin everything for me.
Other games that affected me in this way particularly strongly: Tales of Vesperia, FFIX, Valkyria Chronicles, Dragon Quest VIII, Ace Attorney 3, Radiata Stories, etc.