I'm going to poop out what is sure to be a wildly unpopular idea. We all know the ancient discourse about how hit points (HP) in D&D alternate between physical or "meat" points and luck/fate.
In this hack we focus on HP purely as fate or luck, a timer set between the character and their ultimate doom. We achieve this by simply disallowing healing. Every hit is either just flesh wounds or near misses until the last one. Healing, natural or otherwise, is purely a narrative conceit. Levels would still add HP if anyone manages to eke out a level under this method. It would in fact be the only way.
The draw to this method is that getting to the end of your HP means you are probably nearing the end of your story too, so make it count. Does only the player know, or is the character also aware of their upcoming demise?
I think you will find that the longer you play, the more you will realize that the death of a character is often the best part of their story. Too many players are robbed of the best experience in role-playing by intrusive systems that prolong their lifespans unnaturally. Do you really want to hear about that party of super heroes knocking God on its ass, or the scrappy adventurer who tried to grapple a Grell under the light of a dropped and guttering torch? Look deep in your heart, you know the truth.
Naturally this makes Clerics and healing potions useless. C'est la vie. Leave them out, or make healing potions exceptionally rare.
Enjoy, if you dare.