HP Weariness
A more dangerous variant of Drop Dead Exhaustion and can be paired with Negative HP.
HP Weariness
Whenever you are reduced to 0 hit points, you must immediately make a Constitution Saving Throw with a DC of 10 + 5 x the Number of previous times you've been knocked down since your last Long Rest (min 10). If you fail this save, any excess damage that reduced your HP to 0 instead reduces your Hit Point Maximum by an amount equal to the damage taken.
This Reduction lasts until you finish a Long Rest, unless you are playing with a Variant of the Drict's Resting Rules, in which case it lasts until you finish a Week Rest.
DM Commentary
These rules and subsequent rules regarding stabilization, HP, and health regeneration are meant to represent a more realistic or at least grittier form of 5e that promotes downtime necessary to return wounded or outright close-to-mortally slain individuals. These are often presented in games to place a cap of adventuring capabilities, and to show the slow drag of time as our heroes who walk away bloodied and battled after a gruesome fight, don't pop up fully capable the day after, as good as a great night's rest can be.
Not every adventure necessitates weeks of napping in-game, and not every story benefits from "The Heroes were out cold for five weeks while the world kept turning", but surely places more weight on the importance of good health, and makes the grueling journies across the world and far from places of shelter that much more a risk to consider.
Luck and misfortune surely place a part in whether or not a Character dies from the absurdity presented from every one of these Rules paired together, Drop Dead Exhaustion, HP Weariness, and Negative HP in the same household make for abusive partners, only expected for a truly gritty campaign, and only one should ever be present in a version of adventure that necessitates the themes of the difficulties of adventuring.
That being said, HP Weariness and Negative HP probably tie close together hand-in-hand for the worst couple, though one or the other place a sort of tactical risk, and turn up 5e's more lax version of Character Death into something that may just act as a Meatgrinder instead. A game that relies on these types of rules, probably utilizes less monsters that deal a shit-ton of damage in single hits, especially Dragons whose breath weapons would hit for a character's full HP and then some.
Pretty unfair.
Negative HP
Akin to the gritty HP rules of games like Pathfinder and 3.5, the 5e version of Negative HP is somewhat similar, written here, using the variant of Negative HP changes the text of "Dropping to 0 Hit Points" on pg. 197 of the PHB.
Instant Death
Massive damage can kill you instantly. When damage reduces you 0 hit points and there is damage remaining, you gain a number of special points called negative hit points equal to the remaining damage. Negative hit points are not considered hit points. Negative hit points persist between turns and there is no limit on how many of them you can have at once. If your total number of negative hit points equals or exceeds your hit point maximum, you die.
For example, a cleric with a meximum of 12 hit points currently has 6 hit points. If she takes 14 damage from an attack, she is reduced to 0 hit points, but 8 damage remains. The remaining damage is converted into 8 negative hit points, which are kept between turns. Her unconscious form is hit by a second attack that deals 4 damage. The cleric suffers her failed death saving throws, but also gains another 4 negative hit points. She now has 12 negative hit points, which equals her hit point maximum, so the cleric dies.
(Death Saving Throws remain the same.)
Stabilizing a Creature
If a creature with negative hit points were to regain hit points, the amount of healing instead reduces the creature's negative hit points. If the healing reduces a creature to negative hit points to 0, any excess healing is applied to the creature's hit points.
When a creature is stabilized, through magical or medical means, they no longer make death saving throws, but the amount of negative hit points they have remains until they regain 1 hit point after 1d4 hours.