Have you eliminated hit point inflation from your 5E game? How did you do it? Did it work?
We've done this a couple of times, in a couple of ways. Here's how it all went down.
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Fixed HP: Back in the days of 3rd Edition, we capped everyone's hit points. You started out at 1st level, with a number of hit points equal to your Constitution score. Got an 18 Con? Sweet, you have 18 hit points. Then, at every level-up, you got +1 hit point. And that was that.
This meant that damage had to be scaled back for the monsters, and the party had to become a lot more comfortable with fleeing, using skirmish tactics, surprising their opponent, etc. And at higher levels, I had to make resurrection and reincarnation more widely available (and less expensive). We ended up abandoning this rule at around 9th level; it just wasn't viable for high-level play.
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Wound Meter: In 5E, we experimented with "wounds." Every character and monster in the game could withstand a number of "wounds" equal to their max hit dice. Fighters have d10 hit dice, so they can withstand 10 wounds. Clerics have d8 hit dice, they can withstand 8 wounds. There were tweaks for Legendary monsters and such, but you get the idea.
Then, through the course of the adventure, any time you take any amount of damage for any reason, you take a wound. Fall 10 feet, getting struck by an arrow, or getting blasted by dragon-fire, doesn't matter--it costs 1 wound point. When you run out of wound points, you fall unconscious and start making death save throws. Short rest restores 1 wound point, long rest restores all of them, and cure wounds cures a number of wounds equal to the spell level used. (And this is neither here nor there, but players had a special "desperation move" that they could only do if they had 1 wound point remaining, and rolled a critical hit.)
We've only tested this for a couple of one-shot adventures, and it was a lot of fun. We haven't tried to run it as a campaign, though. It cuts through a lot of the math and makes resource management a bit easier, so it might be good for very young or very new players.
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Anyhoo. That's my two coppers anyway; these options certainly aren't for everyone. As always, if something isn't broken in your game, you shouldn't feel obligated to fix it.