Two series come to mind: Legacy of Kain Series and Dead Rising.
Though very dark and depressing story, Legacy of Kain was one of the few game series I can think of where every game continues to add onto the story and some even revisiting places and times from previous games like a good book series does. The added bonus is the series is made up of two stories.
Blood Omen followed the story of Kain who is killed within the first five minutes and returns to the living world becoming a vampire doing so. His quest for vengeance takes him across the world seeking those that wanted him slain and eventually, his own conquest of the world to stop those very beings and others from destroying it.
Soul Reaver takes place 1000 years after Kain conquers the world, you play Raziel, his favored lieutenant until Kain suddenly betrays him. Raziel left a shade of his former self seeks revenge against his former lord and starts what becomes a world-changing adventure throughout time-and-space.
Both titles tie together in the last game, Defiance where you play as both Kain and Raziel who now work together to stop the one pulling the strings from the shadows of history and hopefully give the world a future worth living.
Dead Rising, on the other hand, is a great story with each individual game. The story and its ending are created through the choices you make per game play. Solving the mysteries behind the zombie outbreaks, trying to save people and fight off psychopaths, all while the clock is ticking makes it intense and rewarding experience.
Overtime mode and the True Ending made Dead Rising 1 for me because just when you think it's over, there's more story to play. All the inner details of that story also made it worthwhile.
In Dead Rising 2, you got all the goodness of Dead Rising 1, but now you have to make sure your daughter doesn't turn into a zombie by giving her a medical shot every day between 7 and 8 AM if you have it. The emotional scenes between Chuck and his daughter were some of the best I've seen. Added onto the fun was its DLC, Case Zero. This further expands the story by going back to before they both arrived in Fortune City. The day I got DR2, I bought the DLC and play through it first before the disc was ever put in the tray. It added a whole new experience for me. I usually don't recommend buying DLC, but you've got to for this game. Also when you beat it, beginning at Level 5 isn't a bad way to start your first playthrough.