Hades! Whenever you die, you get reborn in the "house" of your father Hades. Dying and being reborn is an integral part of this game and is what keeps the story going. You also get to upgrade and unlock weapons that way. Highly recommend this game if you like fastpaced and smartly designed action games!

This is an odd one, but Rimworld.

If your colony is close to collapsing, you have a chance for a "Man in Black" event where a stranger in black comes in and, hopefully, turns it all around.

But what if the MiB doesn't trigger? Hell, what if they're a pacifist pyromaniac with a meth addiction who wandered into a mass of cannibal sex slavers having a rave over the ashes and dies?

Someone will eventually come. It might take in-game years, but eventually, a pawn will come and want to make those ruins home. You can try to rebuild.

Admittedly, it can be quicker to just call it done and roll up a fresh colony over watching the seasons pass, but I like how even a complete loss doesn't mean the story is done.

Atrio: The Dark Wild - has you control a clone with a limited life span. When you die and resume from a new clone, the old clone corpse is lying around and you can harvest it for parts necessary to continue the story.

Sifu - when you "die" your character ages and gets stronger before trying again.

Karateka - plays a lot like a regular game with lives, but it's not the same life. Every time you have to resume from a new life, it's a different person attempting to get to the end.

Shadow of Mordor - when you are killed by an orc, you resurrect from a spirit. The orc, however, gets high-fives from all his mates and gets promoted, plus some new skills. Next time you see him he will call you out.

Hades - the entire story is based around you repeatedly failing and dying.

Super Meat Boy - well basically you die and restart, but when you finally beat the level, you get an instant replay with all your failed attempts simultaneously playing on top of it. The effect is more glorious the more you struggled to beat the level.

Planescape: Torment is an old PC RPG similar to Baldur’s Gate 1&2. Your character recovers from death in the morgue (which is where the game starts) and occasionally it will trigger memories in your character, who has amnesia of sorts.

Kenshi. Though usually that means that your corpse was found by slavers, nursed back to health, and its up to you to find replacement limbs and then crawl/hobble/run away from the camp when no one is looking

Hylics and Cruelty Squad both spin death.

In Hylics 1 & 2, dying causes you to wake up in the afterlife where you can take the chunks of meat you get from enemies and put it into a meat grinder to increase your max HP.

In Cruelty Squad dying is just a consequence of living. It happens sometimes. Dying severs your divine light, making the game easier but closing some paths to you. Additionally, if you die too often, you'll find power in misery, making the game easier again and allowing you to consume bodies to restore 1hp each. This is particularly advantageous because eating bodies dismembers the corpse, allowing you to harvest its organs without having to chase them around (most other ways of gibbing corpses tends to send organs flying). Additionally, you can get death surgery, allowing you to pass through some areas and use a few weapons that were previously too dangerous for you to access. Death surgery also allows you to wall jump.

Outward! A relatively low budget but very enjoyable action RPG with surprisingly non-annoying and actually fun survival elements.
Whenever you die in Outward, a random "defeat scenario" occurs. Sometimes you wake up rescued by a stranger, sometimes someone brought you to the nearby town. And sometimes you wake up as a prisoner in a local thug camp and need to figure out how to escape.

I made some heavy mistakes in Act 1 of Baldurs Gate 3 and the game is still continuing, now with fewer options for characters that I can include in my party, because one died permanently, one left and one even refused to join.

If that's want you meant?

In Thief 3: Deadly Shadows, when a city guard kills you for the first time you get sent to prison instead of dying. It's a cool "bonus" level many players can miss because city guards are the easiest enemies.

I think it would be a cool idea if they gave other factions (Pagans and Builders) their own bonus levels

Most of the comments focus on death states, as far as I recall you can totally beat TES 3 Morrowind after an essential npc dies. The game will tell you it's doomed and will prompt you to load a save, but you are largely able to continue, just have to live with the consequences, it might be a pain to do or rely on cheese, but apparently technically possible.

Antichamber was pretty good for this. You would accidentally fall off a bridge or something and expect a game over, only to find an entirely new area to explore. There were no failure states as far as I remember.

Rogue Legacy! You are a knight invading an evil wizard's castle. When you die, your children take up your mantle and try again.

Dying means you get to try again with a descendant that has different quirks, like "being left-handed" or "dwarfism"

Besides all the roguelikes people mentioned, Omikron: The Nomad Soul from Quantic Dream has you possess a different body each time you die, which comes with different conditions. The idea was then reworked much more extensively for Watch Dogs: Legion, where you play as a whole resistance movement you can expand via recruitment and jump to a different member upon death.

There’s a PC game called Ctrl Alt Ego (Steam link) where you play as a disembodied conscience that can project itself into - and control - different entities in the game.

When your current host is destroyed you just become disembodied again and can project yourself into another nearby entity (even the enemy that destroyed your host, in some circumstances). It’s quite a unique concept and almost completely removes the need to quick save/quick load.

If you’re into Immersive Sim games then I would highly recommend it - Stands alongside Prey and System Shock 2 IMO.

Disco Elysium is probably the best implementation of the 'Fail Forward' ideology I've seen in a game - not 'Game Over' per se, because running out of Health or Morale will give you a game over, along with some nonstandard endings, but failing important story-related checks doesn't lock you out of the story, you're just encouraged to go explore other parts of the world - raising the skill associated with the check you failed opens it up again, and certain objects, thoughts, or interactions can also open them up again. In the same vein, failing noncritical checks can often lead to more interesting and/or advantageous outcomes than succeeding. As an example:

Project Zomboid is less narrative than what you're looking for, I think, but when you start again in the same world you can find your previous character as a zombie.

All of the Grand Theft Auto games have you respawn outside of a police station or a hospital.

I'm not sure if it qualifies exactly as what you're describing, but Metal Gear Solid 2 had a moment where they subvert the game over screen. At some point in the fight a game over screen comes up but it's full of typos like "fission mailed" instead of "mission failed" and there's a small window in the top-left where the fight is still on-going.

Also, notably, all the soulsbourne games kind of subvert the player's death by making it basically required to continue most of the time.

I've only played through it twice, but iirc Detroit: Become Human is like this. Even if you have a main, playable character die, the story just keeps going, and there a good amount of paths to take depending on the choices (and QTEs, bleh) that you make for the characters. On my second playthrough, I may or may not have repeatedly and purposely got a character killed just to see how much it would affect an certain NPC. 😬

Katana ZERO. The fact that your character can fail and "die" and yet be able to control the flow of time to return and try again is not only contextualized through the game's lore and your character's usage of a drug, but becomes basically the entire story by the end of it. Brilliant game.

Odd Giants is based on the old MMO Glitch, and in that, when your character succumbs to empty stamina, you go to the underworld to recover. It's a truly special game.

In the puzzle platformer Braid you can always rewind time, so any failure or minor mistake can be corrected by rewinding a little bit. Technically there is a fail state where you can die, but rewinding is such a basic mechanic, going back feels seamless.

In Chernobylite, when you die (either by getting killed in specific circumstances or by committing suicide in a special device), you get a chance to alter the past by changing decisions you made in the game, which will end up in changing the story and a whole lot of things like companions’ attitudes, weapons at your disposal etc.

They're oldies, but both of the Soul Reaver games and the Raziel parts of LoK: Defiance have a two tiered death system. (Minor spoilers going forward, but I'm talking spoilers for the opening cutscene from Soul Reaver, so I'd argue not real spoilers.) See, after having his physical vampire form thrown into the well of souls and falling through an endless abyss for several hundred years, Raziel exists first as a structurally sound spirit in a world where most things don't maintain any resemblance to their corporeal forms after death. After being salvaged from the well of souls and set on a quest, he gains the power to create a corporeal form that resembles wretched wraith he has become in spirit. But that new corpse isn't him, and destroying it doesn't kill him. It just releases him back to the spirit realm, where he can regain his strength and manifest a new corporeal form. You can be killed in the spirit realm and sent back to a checkpoint, but the spirit realm is by and large far less dangerous than the physical. Very few threats can follow you, and the soul scavengers who populate most the spirit realm are little more than fodder for a creature like Raziel to slay and eat at his leisure. So you regain your strength, find a nexus between the worlds, and manifest a new body. It's probably my favorite unconventional handling of death or failure in a game because of its comprehensive and essential connections to the story and lore of the series. Raziel is the beating black heart of the mythos of the series.

Death stranding

Pathologic 2, it's a Russian horror game. There are real, permanent repercussions for dying and loading a save won't help you get rid of them. Sadly saying too much is a huge spoiler, but there's actually a storyline that's revealed with each death. Your deaths also affect the physical world.