We start as kids
I'd like a game where we start playing as children dealing with something fantastic/mysterious/horrifying. After some time, we jump forward 20 years and refocus on our characters as adults.
This is actually two games, one as kids and one as adults, the second half basically dealing with the fallout of the first and looking back at the children's world through adult eyes.
Ideally the "children" portion wouldn't be just a prelude, but a whole self-contained game of its own. It might cover the events of a single year or a single summer and then jump ahead to adulthood... or we might skip and jump from childhood to adulthood gradually. There could be a childhood "adventure", then a young teen adventure, a teen adventure, then an "adult" phase. We might continue jumping ahead or just treat the "adult" era as ongoing.
Possible permutations:
1. A world of darkness game. It could be a Vampire/Werewolf/Mage game, though there should be a convincing plot reason as why this group of friends all happened to be embraced/awaken/etc. A Mortals game would be easier to justify.
2. A superhero game. Maybe we were normal kids growing up in a world with superheroes who gain powers -- somehow -- as adults. Or maybe we got our origins as kids, and were a kid-hero team, then we deal with how that affects us, and how being an adult hero is more serious/deadly/less glamorous. I would vastly prefer a postmodern sort of supers game rather than a gold/silver age sort of thing, though maybe when you're a kid you THINK it's all just like the comics.
3. A fantasy game. We grew up in the same village or castle or town, having "adventures" as kids that are little more than play, but when adulthood comes we're forced to become adventurers for real.
4.Sci fi game. The kids grow up on a starship; when we're adults we can go to Space Academy and serve on a ship of our own.
5. Sci fi game. Like above, except we grow up on some backwater planet. When we grow up we can buy or finance our own ship and get off this miserable rockball.
Etc.
Core to this is the ability to portray children accurately, both mechanically and as characters. I'd prefer a system that supported this: GURPS, nWoD, Fate, maybe Mutants and Masterminds 3e or Savage Worlds, and a GM/player group who gets what it's like to be a kid, and not the stereotype of a kid.
Unless it's a situation like kid-superheroes, the game should start out fairly low-level. Power level might ramp up after we become adults, but this should be part of the game we play out.
Inspirations: It, Star Trek: TNG, Harry Potter, Stand By Me, etc.