The Mass Effect Series.
Stick to the same story and same level of cadence, but bring it all up to next generation graphics, improve combat, and no thermal clips. Second game needs to be affected by the first in more ways, needs to be more relevant to the overall story (No Human Reaper nonsense) and needs to keep the galaxy as the #1 threat instead of just humanity. Third game will need to go through a complete rewrite. I know that there are a lot of "95% of it was awesome until the ending!", but there are so many elements outside of bugs that just don't compliment the overall story.
If anime companies can redo their own shows (Ghost in the Shell, Evangelion) and games (Biohazard), I'd like Mass Effect to try and do the same. Only this time, have writers stick with the entire series and actually commit to plot points that the each game spends time establishing. And of course, REAL multiple endings that are so broad, you cannot possibly get half of them if you select "new game" in part 3. Good jumping in point be danged.