Existential Horror
I'd like to run a cosmic horror game. Not like Lovecraft cosmic horror, but existential "everything we think we know is a lie" sort of personal psychological terror. A game where the players are willing to explore and examine their characters' reactions to assaults upon what they believe reality is.
Example: Typical RPG - Godzilla is scary because he can crush or eat you, and there's nothing you can do to stop him. You are insignificant.
Existential Horror - Godzill is scary because it violates the way biology is supposed to work. It's violence against reality that calls into question causality and every assumption we have; if monsters exist, what else is science wrong about? Also, what about the square-cube law?
Or you can pick other typical RPG tropes that violate our notion of the way the world "should" be. Magic, monsters, etc. Not "unexplained by science", but "violates what we believe to be possible."
There will be typical RPG adventuring as well - problem solving, social interaction, and combat - but the deep internal wrestling with violated worldviews is something I want to emphasize, and that's entirely up to the players.
If this kind of deep RP seems interesting to you, I've got two ideas.
1. The Anomaly. Science Fiction, set in the future. Players are scientists sent into deep space to examine a scientific anomaly that turns out to be much more than a simple curiosity; something with dangerous and maddening implications. Players might also play ship crew.
2. The Dungeon. Heroic Fantasy on the surface. Players are Seasoned AdventurersTM sent on a quest that turns out to be more dangerous than any dragon or evil wizard, something that even the wisest of sages would be driven mad to comprehend.
I could run either game (or both; they're obviously different). Both would be run in a black-box home-brew system.