I think every game is a different case.
If i were to say franchises as a whole... probably the Tales series. I would totally buy new Suikoden's, Legacy of Kain's and Grandia's on day one if they were still making them. I guess Mass effect is a day one affair aswell.
If we go by game from this year and last year off the top of my head:
- Kingdom Hearts 3D
- Fire Emblem Awakening
- Pokemon Y
- Ys memories of celceta
- Tales of the Abyss and Tales of Xillia
- Dynasty Warriors 8
- God of War ascension (this one is a sort of because it came bundled with my PS3. I enjoy the series but i dont usually buy them day 1)
- Soul Sacrifice
- Ace attourney 5