If a game designer wants an NPC to be invulnerable, they'd better make sure that it truly is.
The Lord British Postulate was coined in thisWoW Insider post, and is the cornerstone of the creativity and ingenuity of the more cunning player. In short, if a NPC cannot be killed through normal means, then there must be some abnormal means to do the job. Quite often, if a character's weapons fail to do the duty, then the task falls upon other NPCs, world objects, or the environment to assassinate the relevant target with some prodding from the player characters. The Lord British Postulate is not restricted to MMORPGs, but can be applied elsewhere: CRPGs, FPSes, even Tabletop RPGs.
The name references the fact that there is a way to murder Lord British in nearly every Ultima game, famously including Ultima Online, where Lord British was killed by a fire spell because his invulnerability flags hadn't been set properly after a server crash. That said, the Lord British Postulate has less grounding in reality than one would assume. A great deal of NPCs are free to walk around unmolested, safe in their invulnerability. However, the more important the NPC is, the more likely it will be the target of a creative assassination attempt. A king is a bigger target than the common baker and much more satisfying to dispatch.
This trope can be an applied form of Loophole Abuse with how you go about killing Lord British (or any other normally unkillable NPC).
Genre Savvy developers will code in what happens if you do manage to kill them.
Contrast with Invulnerable Civilians, which simply can't be killed. A Hopeless Boss Fight or Invincible Minor Minion is seen as a challenge to these people. Can lead to the game being Unwinnable by Insanity if the loss of the NPC screws you over. This has nothing to do with the other ship from Gradius.
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In Soul Calibur III, the subboss Ceres is supposed to be unbeatable (you are told to endure the battle until time runs out without getting killed), but it is possible to beat Ceres if you attack very quickly and break his guard. Probably because the game designers didn't intend you to beat him, nothing special happens after you do.
Fighters Destiny and its sequel have a minigame where the objective is to stay in the ring as long as possible against a seemingly invincible bull. In reality, the bull can fall off the stage, though if this happens the round ends as if you fell.
In Super Smash Bros. Brawl, the supposedly invincible Wario-Man, as well as characters using a Starman, can both be KO'd in the conventional fashion by chucking a Team Healer item at them while they're at a high enough percentage and not on your team. On a similar note, while Giga Bowser is completely Immune to Flinching, he can still be star KO'd if he's hit while he's off the top of the screen.
In stamina matches, both characters still take damage; defeating while in these forms will cause them to revert.
First Person Shooters
In Golden Eye 1997, a glitch involving using two controllers at once allows the player to fire on characters during cutscenes from the camera's perspective, instantly killing any that are hit. This includes Baron Samedi during his The End... Or Is It? moment. This has no gameplay impact whatsoever, though.
The Jedi Heroes in the first Star Wars Battlefront game were invulnerable to normal weapons; they would deflect blasters and simply get knocked down by splash damage. It is possible to kill them by using grenades or heavy weapons to knock them into environmental hazards: off the edge of Cloud City, into the Sarlacc pit, or into the Carbonite Freezing chamber. It is also possible to kill them by landing aircraft on them or ramming them them with a speeder bike.
An even easier method is to get them between a few turrets firing at the right rate. The flinch effect will keep them immobilized long enough for you to take them out. Sadly, they'd just respawn like Agents in The Matrix.
You could also do this by using a vehicle to push said Jedi off the playable area; though this will kill you eventually, it'll also kill the Jedi, which will (for whatever reason) not damage your vehicle whatsoever. This works great with the Galactic Republic's tank.
Deus Ex has plenty of invincible NPCs, but many can be killed due to a collision bug. Get one to stand on the edge of a piece of furniture, then push the furniture against a wall. They will pass partially through the wall and then die instantly as the game seems to think they've been crushed.
In Deus Ex: Invisible War, important NPCs aren't coded to be invincible, but are simply placed in areas where you can't use your weapons, or speak with you from behind bulletproof glass. One such important NPC, Billie Adams, can be seen at the beginning of the game running into an inaccessible booth before speaking with you. It's possible to kill her before she gets into the booth. This removes her from the game, but in later encounters there will be gaps in the dialogue where she was supposed to be, because the game was never programmed to deal with her not being alive.
Interestingly, the engine does support flagging NPCs as unkillable — though any NPC flagged as such is in a place where weapons are disabled, with one obscure exception, so most players will not notice. Why Billie is not flagged this way is unclear.
One side-quest has a canopy with a weapon and a scientist. Open the canopy with a button press and the scientist gets locked behind an indestructable door. Kill the scientist and the canopy locks down instead, denying you the weapon. To achieve both, place a metal box so that the door cannot close, open the canopy, get the weapon, and then head for the scientist. Get a melee weapon and hack away (which will really take a while due to obscene amounts of health). For your effort, you get chided from one faction for helping the other, but later still get their award as well as the other faction's bonus.
Prior to being patched out, it was possible to kill Eli Vance in Half-Life 2. The player's attacks don't hurt him at all, but if you grabbed a sentry gun from earlier in the level, carried it all the way to Eli, and set it down facing him, it would quickly blow him away.
Another fun thing to do? Start a new game, get God mode going, get the HEV suit, THEN use Impulse 101 running. Get ammo and the crossbow, and follow the steps up to the teleport cutscene. As you are transporting from place to place, you can headshot Breen either place; when he first sees you and you fade out, or when he rounds his desk.
The physics engine can be exploited to kill certain otherwise-invulnerable NPCs using the Gravity Gun and nearby physics objects, and done at the proper times, you can avoid triggering a Nonstandard Game Over where you would otherwise receive one. Observe.
Bioshock features an invincible Splicer named Rosenote She's the one that kills Johnny in the very beginning while you're trapped in the Bathysphere that can only normally killed in her final appearance, but it is possible to kill her sooner on Easy if you've stocked up on anti personnel machine gun rounds.
Lord British, of course, from the Ultima series, who just happens to be one of Richard Garriott's two in-game Author Avatars. The postulate itself stems from a time during the Ultima Online beta test where, following a crash, Lord British' invulnerability flag was unknowingly turned off, and a player killed him with a hastily-stolen firefield spell on the spur of the moment during a public gathering. The guards moved in, the players fought back, and the whole thing turned into an in-game riot.
There was an event during the early weeks of Tabula Rasa that was about killing General British. Well, clones of him, actually; there were dozens of them.
Most NPCs in World of Warcraft are actually killable to players of the opposing faction, including racial leaders such as Thrall, Jaina Proudmoore, and the like. However, most NPCs in the neutral city of Shattrath should be invulnerable, as combat from players is forbidden there... unless, of course, one pulls an enemy from outside of the city into it and lets them run loose on the population. The main NPC of the city, a massive naaru named A'dal, was killed in this way several times until it was buffed to nigh-unkillable levels. It seems that the original A'dal had very few HP.
It's still possible to kill A'dal with a quest mob that becomes immune to everything at 50% health (you need a quest item to kill him). It just takes a very very very long time. In theory, it would also be possible to kite said mob to A'dal, then kite him away, effectively kiting A'dal itself (and every guard in the city). There is a rumor of this being done, but there are no videos to back it up.
The pre-launch event of the second expansion involved an infection that would turn player characters into zombies that were hostile towards everything but each other. You can use this opportunity to kill NPCs that were otherwise unattackable to both factions.
In the third expansion, one guild managed to kill Tirion Fordring by luring a mini-boss from earlier in the dungeon to him.
Invoked with the Whale Shark. An immense boss mob with no loot of any kind and the ability to one-shot anybody that attacks it, Blizzard had to add an achievement for killing it because so many people wanted to.
Some players expand this to include other players when not in PvP situations that allow deaths. Duels end with the other player having one hitpoint, but using knockbacks or mind control during the duel, it is possible to kill via fall damage.
Perhaps the worst incident of this behavior was the Corrupted Blood Plague. Players discovered they could bring a highly-contagious disease to their home cities, where it would promptly spread and kill characters who were ordinarily protected from other players.
Early in the Mists of Pandaria beta, there was a bugged version of Chi Wave. The spell is supposed to bounce between PCs and NPCs, healing if friendly, damaging if an enemy. Earlier versions bugged so it would damage all NPCs.
In Guild Wars, a well-placed herd of Necromancer minions can sometimes kill exactly the wrong person during a mission cutscene.
The Eye of the North expansion is far more explicit with this. If an allied NPC is killed, it will either return hale and hearty on the next dungeon level or, more commonly, wait until its aggro circle is clear before standing back up, dusting itself off, and running back to your location. Quite a few NPCs will rubberband back up to about 30% if their health drops below this threshold, even if the damage they receive puts them deep into negative health. Of course, it was possible for decorative player minipets and certain resurrection NPCs to be accidentally killed by rolling ice boulders when they shouldn't, but this was quickly patched.
One month, it was discovered that a specific buff could be used to spawn minions in some outposts by gaming the system slightly. As players do not have access to skills in outposts, the minions would quickly kill them, and their corpses would spawn more minions.
Kerafyrm the Sleeper, the Sealed Evil in a Can from the "end" of the original EverQuest. He was supposed to be unkillable, but on one server he was eventually taken down by a group of over 200 allied players in a battle that lasted three hours.
Also a particularly lasting example of Lost Forever — Kerafyrm could only be awakened once per entire Server, meaning a failed attempt prevented all others on the server from ever being able to do the event.
The issue here is that Kerafyrm was specifically designed to appear to be killable, so people would trip the sad story event caused by the failure. People were supposed to think they could win, and be on the end of a giant Player Punch when they inevitably lose. They couldn't make him actually unkillable, because once the first group to wake him found out, no one else would do it, because there is no benefit from trying and failing.
And while we're on the subject, Verant had to make characters used in GM events untargetable specifically because of this problem.
Yohoho! Puzzle Pirates has El Pollo Diablo, the black ship that appears if players attack ships weaker than themselves too often. It can't be shot, and it's crewed by (noq) 150 skeletons, but it has been defeated multiple times, each matched by a severe upgrade by the developers...
... Except the one time they accidentally botched the ship to be crewed by just 8 skeletons.
Further, the hardest defeat so far, 100 skeletons versus a full Grand Frigate of top players, had one of the Game Masters fighting on the Black Ship, for full Lord British Postulate goodness, no less!
They are mostly beaten through boarding and stalling.
In the MMORPG Runescape, one update brought a seemingly invincible enemy called the Vyrewatch. They've got a combat level, so they must be killable somehow, right? Someone went to the trouble of getting one to attack him while a group of friends dropped him rings of recoil and massive amounts of food. Eventually, the Vyrewatch died, with no death animation and no drops, proving that they really were intended to be invincible. However, they have been made killable several years ago.
Final Fantasy XI has the ostensibly killable Absolute Virtue, which does in fact have death animations, text, and very valuable (and exclusive) drops... but every time it's been killed so far, Square Enix respond with something along the lines of "No, that's not how you kill it" and patch it. Sometimes they ban the players who killed it, too.
Also notable is the actually newsworthy superboss Pandemonium Warden. When it was first patched into the game, it was deemed truly unkillable, and one linkshell group spent 18 consecutive hours in a single fight against it and could not continue. Since then, SE patched it to make it actually very possible to kill, but it is still comparable to Absolute Virtue in difficulty.
Part of SE's "fix" was to cause Pandemonium Warden and Absolute Virtue to despawn after so long, preventing the 18 hour long fights that nearly hospitalized some. PW will despawn two hours after being spawned. Absolute Virtue only spawns after killing the Jailer of Love NM, so AV despawns two hours after the Jailer of Love is spawned. This is particularly troublesome as one of the greatest obstacles to killing AV is its insane health regeneration, which can be mitigated by killing a lot of Jo L's pets. This basically means to have any shot of dealing enough damage to override the health regeneration, you have to keep Jo L alive so long you have less than 90 minutes to kill AV.
Note that Absolute Virtue originally came about when the level cap was 75. The level cap is now 99, and it's still insanely hard to kill, though much more possible than previously. No, they didn't make it harder at any point.
City of Heroes has the Hamidon, a giant blob monster, which was similarly very difficult to kill. For the first year or so, players kept coming up with strategies only to have the Devs change the rules. Eventually the players found an "acceptable" strategy and raiding began in earnest.
CoH also has Riechsman, who's supposed to be completely invincible... however, when he first came out, he was still susceptible to Sleep abilities, which, while not killing him, completely mitigated him. Also, later on in the same set of missions, you get the means to defeat him... which you can keep, then do the story over again, and use the same weapons to defeat him when you're supposed to be helpless against him!
The Rikti Invasion events in City of Heroes feature dropships, which the dev team made virtually unkillable, with oodles of hit points, powerful attacks, and a tendency to simply fly away and escape before anybody can kill it, and which give no rewards for defeat, specifically because they were not intended to be defeated. When the Issue that introduced the Invasion events was in beta, a group of players took this as a challenge, banded together, and killed nine of them. The devs then made them even tougher, but still not unkillable.
On a smaller scale, the "Magic Man" arc has the eponymous character who, during your battle against him, uses Time Travel to heal himself repeatedly when his health gets low. Mechanically, this is implemented as a new copy of the NPC running in while the low-health NPC copy gets flagged as invisible and untargetable, but not untouchable. Area-effect attacks and pets can still hit the old copy, and you can get XP and drops for defeating him as many as five times.
The finale of the first chapter of the villain "Who Will Die" arc ends with five high-level, high-rank hero NPCs arriving to drive the player character off. These NPCs are twenty levels above than the player characters (in a game where a seven-level difference is considered insurmountable), but have no special protection. Player speculation is that concentrating confusion attacks on the healer NPC would get her to wipe out the other heroes, but nobody has claimed to have actually done so, and that still leaves the problem of defeating an incredibly tough NPC with self-healing abilities.
In Champions Online, Justiciar in Canada is positioned dangerously close to a pair of explosive barrels which respawn. So long as no one's Nemesis minions show up to draw him into combat (and thus full heal him when he returns to his default position), a persistent player can kill him with the explosive damage from the barrels.
Another instance of this were statues in one of the endgame lairs. These statues were immune to all damage types, and shot lasers out of their eyes when approached, which the player had to use a mirror to reflect back at the statue to destroy it to progress through the lair. However, characters who had Parry could actually guard to penetrate the statue's immunity to normal damage, allowing players to bypass complex mirror puzzles by just waiting in front of statues while blocking.
In EVE Online, CONCORD serves as the police force in high security systems and will attack players who perform certain illegal acts, such as attacking other players without provocation, attacking a CONCORD ship, entering highsec space with a low security rating, etc. If a player commits such an act, an endless stream of CONCORD ships will very quickly enter the system and destroy the player's ship. Due to their weapons and equipment, most Concordokkened players don't even have a chance to fire a shot at them. There was one case, however, called the Yulai Incident, where a well-organized and well-prepared gang was actually able to hold off CONCORD for over two hours, killing over a hundred people in a supposedly safe system.note For the interested, they were able to pull this off because their shield and armor boosting ships didn't get flagged by CONCORD. That was considered a bug and was quickly fixed. CONCORD has since been upgraded (and is so every time anybody tries to tank them) and everyone involved was banned (tanking CONCORD is considered an exploit and bannable offense), but the Yulai Incident lives on in legend.
Dungeons & Dragons Online has a number of "invincible" monsters, usually seen but not directly encounted in inaccessible areas of dungeons. Most of them have enormous HP reserves and full health regeneration. This hasn't stopped players from using high-level characters with powerful ranged weapons/attacks to attack Aussircaex, a level 32 dragon (player cap is 25 currently). However, this is usually true only in the first dungeon they're encountered in - most are quest-end bosses in separate adventures, since Turbine is aware that players are going to attack anything targetable.
The proto-MORPG AberMUD featured a dragon which had 4x the hit points possible for a level capped player, and did approximately the same damage per round as the best weapon in the game. It was, however, a one-hit kill if you had a specific unique (as in, one copy in the entire game world at a time) weapon equipped. Naturally, a few people took this as not a puzzle to be solved but a challenge.
It also had a unicorn with 4x the hit points of the dragon. This wasn't even in the game proper; it was intended as a GM's "pet," but the mechanics allowed players in certain circumstances to attempt to summon any NPC in the game. Players who thought the dragon wasn't enough of a challenge would have a go at the unicorn. If superbly equipped, it was possible to win this fight, though the odds weren't terribly good. Later GMs learned from this and, if they bothered to create a pet, created one that was either weak enough not to be worth killing for bragging rights, or with hundreds of times as many HP as players and attacks strong enough to kill level-capped players with a single hit.
Phanto in Super Mario Bros. 2 can be killed by exploiting a glitch. If you have a Toadstool Block (one of those mushroom shaped things you usually need to climb on), you can throw it at the key making it jump in the air. If you do it about a dozen times, the key stops reacting, and instead glows and makes an impact sound. You can see it in action here.
It turns out that Phanto did not appear in that video because the longer you aren't holding the key, the longer he takes to appear. Why the key did what it did is anyone's guess.
The Angry Video Game Nerd also found out a way to kill Phanto through a similar glitch. You had to get four cherries, pick four large vegetables, pick the fifth to get a clock to stop time, get another cherry to get an Invincibility Star, and while you're invincible, slam Phanto. However this is a more specific version.
A staple enemy in the Kirby series is Shotzo, a little cannon that shoots cannonballs at Kirby. Normally, Shotzos are invulnerable, and must be avoided, or dropped into a pit. However, Shotzos sometimes come in from the sky, floating down with Parasols, which are far from invulnerable (destroyed on contact with anything faintly Kirby related). Normally, if Kirby sucks up the Parasol, it'll simply come off; however, the Throw ability found in some of the games will pick up both the Parasol and the Shotzo, and when said Shotzo hits a wall, it pops like any other enemy.
In Rogue Legacy, traps that are immune to damage from your weapons (bouncing spike balls, spring-loaded floor spikes, and wall-mounted fireball shooters) can be killed using Retribution runes (which return a percentage of damage taken to the enemy that causes it).
In Jak II the waters around Haven City are patrolled by a sentry bot which will kill the player with one shot if they move too far out. However, Dark Jak can destroy it with a dark bomb, although it may still get a shot off. It won't respawn, but you can only move so far before you fall off the world.
39th Slayer in DROD: Journey to Rooted Hold. You're not meant to be able to defeat him until the final level, but the fans have discovered ways to kill him in most rooms that he appears in. (He always comes back....)
Command & Conquer Tiberian Sun: Firestorm has two of those. The CABAL Core Defender (a boss actually) who should be unbeatable except when abusing his Super Drowning Skills or using stealthed One-Hit Kill capable WALLS, and the super-charged veinhole monster from a GDI mission, which will die when you fire long enough on it. ("Long enough" being a long time.)
In Warcraft III, it is possible to destroy the undead camp in the final mission, which completely stops the attack until the scripted event begins where Archimonde scales the mountain personally.
A mission on the main path of Starcraft II involves racing against the Zerg in an attack on a nearby Protoss base to grab an artifact. If you put off that mission and do many of the side missions and research and upgrade your units, it is possible to destroy the Zerg base. Although the AI cheats with regards to the resources and larvae it has to spend, all attacking units it creates outside the drop pods are built normally, and destroying the base makes it impossible for any units (other than the aforementioned drop pods) to be created.
Any mission against the Tal'darim in Wings of Liberty that doesn't involve wiping out their forces as a primary objective can still be completed by wiping out said forces. Matt Horner will indicate that the Tal'darim are in full retreat and the mission will end successfully.
In most of of the Ultima single-player RPGs, Lord British is not invulnerable — just very, very, very tough, with limited invulnerability to all but certain weapons. In later installments the developers caved to player's persistence in finding ways to kill him, and started adding easter-egg murder methods on purpose.
This article details all of the ways that Lord British can be turned into a horse and shipped to Nebraska in the Ultima series.
It's a feature of one of the expansions of Ultima VII that by using the mythical Black Sword, you can kill Lord British. This is actually one of its features — it can instakill almost anything. Strangely, this doesn't include the final boss, which was lampshaded by Spoony:
"So I can kill Lord British and make the game unwinnable, but killing the villain (which is logical) I can't-... because."
Lord British didn't appear in Ultima VIII: Pagan, but there was Beren, the town sorcerer, who punished any misdeeds the Avatar committed by blowing him up. He couldn't be killed by normal means, but that didn't stop some players from exploiting the game's Super Drowning Skills against him.
In Ultima Underworld 2, it is possible, though extremely difficult, to kill NPCs by pushing them into water if there is a body of it available. This can be used to best effect against Dorstag, whom you are supposed to obtain a quest item from through combat or other difficult means. Though most of his plunderable inventory disappears beneath the water with him, the quest item actually stays on the surface for you to pick up. You can kill virtually all npcs by attacking them too, but drowning them avoids retribution.
Both Underworld games also have the secret spell Armageddon, which kills everyone when it is cast.
The exceptions are the inhabitants of Castle British in Underworld 2, including Lord British. They have unlimited health, there isn't any water available, and high-level spells like Armageddon don't work in Britannia because of the blackrock gem encasing the castle.
Parodied in this short film, where the Avatar attempts to kill Lord British (played by Garriott himself), only to find that swords, axes and even a grenade launcher do absolutely nothing to him.
At least in the German version of Baldur's Gate, children are supposed to be invincible, possibly out of fear of Media Watchdogs. This can be annoying if they turned hostile to the players for whatever reason. However, if you polymorph them into squirrels by magical spell or wand, you can kill the squirrels - and watch them polymorph back into dead children.
Drizzt can actually be killed in a fight, though, in the first game, not a fair one; you could lure him out onto a peninsula in the middle of a lake, then have your characters plink away at him with arrows for twenty minutes or so while he strains vainly in their direction. However, a later patch gave him the ability to go around the lake.
Drizzt can also be killed by a low level party using save-reload exploiting to pick his pockets. A low-level character has only a slight chance of succeeding, but by saving after each success and reloading after every failure you can steal his scimitars because they aren't equipped unless he turns aggro. Equip them quickly on your favorite party members and you can kill him, although his bare-handed attacks can still explode low-level characters, so be careful. It also counts as an evil act.
Spamming summon spells with the Wand of Summoning is the classic way to kill Drizzt.
Very rarely, in the initial build of BG 1, Gorion would kill the armored figure during the ambush encounter, causing the game to lock-up and crash (though it required his spells to hit for 85+% of their possible damage each time, and he needed to kill the ogre on his first spell), the Armored Figure's hp was buffed in the following patch.
Elminster is very killable, but it's EXTREMELY difficult since he has -30 AC (only a 5% chance of hitting him since only a natural 20 will work), 1800 hp, and will despawn if he can talk to you which he constantly tries to do. An attempt requires boots of speed and the throwing axe +2, and managing to avoid him at his first encounter so you can come back later. It takes about 3 hours of running a circle chucking the returning axe at him. He awards 26000 xp.
A good number of characters that are supposed to be unkillable (for example Ulraunt from Candlekeep) rely on one or more of the following: having their HP boosted to absurd amounts (several hundreds in Ulrant's case), making them invulnerable to attacks, and giving you only a very limited time to interact with them. However, a player can delay the event interruption by immobilizing the NPC that is supposed to start it and bypass the invulnerability with Critical Hits (if the invulnerability is combined with boosted HP then this can take hours but is still doable). The creators of the game anticipated that something like that may happen and created a special NPC called Biff the Understudy who automatically appears to say the dead person's lines in their place.
The Elder Scrolls games were notorious for this, especially Morrowind, where everyone and everything was killable. Including plot-requirement NPCs.
It is actually possible to finish Morrowind's main quest even if all "plot critical" characters but one are dead, but doing it without being told how in advance requires incredible luck.
Which may have spurred Bethesda to make pretty much every "important" NPC in Oblivion unkillable. If they "die", they drop to the floor unconscious and rise again when the coast is clear. With very little health. And often the coast isn't that clear. Plug-ins remove this invulnerability, as do certain console commands.
In some cases, you can simply wait until they're no longer relevant to the plot, in which case they're fair game. Other NPCs never have their 'essential' flags expire.
It's also possible to use the Sneak skill to plant an item in their inventory, such as a ring, that does constant damage while worn. They put on the item, and it'll keep their HP at zero even when the "important" flag resurrects them.
Under normal circumstances, Mehrunes Dagon, when he appears at the end of the main quest, is impossible to kill; hitting him with weapons will only stagger him. Except if you use the Wabbajack; although it won't trigger Involuntary Transformations on him, will remove his invulnerability, making it possible to kill him. He has no death animation, though, so when he dies, he just melts into a weird lump of Dagon-colored goop.
Or you can use a weapon enchantment combination of Drain Health 100 and Weakness to Magic 100, each for 6ish seconds, to do over 25500 damage in 8 blows of a melee weapon (100+200+400+800 etc etc), including one blow of 12800 before the 6 seconds of Weakness to Magic runs out. Dagon has 10000 health which regenerates every second. The eighth blow will do just enough to reduce him to a reddish stain on the city floor.
The Fallout series is pretty creative with this, even allowing you to passively overdose NPCs with various harmful drugs.
The Overseer in Fallout 1 is the only character you can't kill (in a game where you can kill everyone in the entire game,) as he has infinite health and will kill you in one hit...until at the end of the game, when he finally steps out of his death chair and becomes vulnerable (in fact, if you have the Bloody Mess trait, the Berserker title, or have low karma, you'll kill the Overseer automatically.)
In Fallout 2 you can kill the children who pickpocket you without gaining the Karma title "Child Killer" by unloading your entire inventory save for armed explosives. That'll show the little bastards. Another way of killing people without being held responsible for it is to pickpocket THEM and use the interface to move an armed explosive to their inventory, then leave the area and wait. Once you return after waiting long enough, you'll find a dead NPC and nobody's blaming you. Clearly it was a freak case of spontaneous detonation.
In Fallout 3 there's even an Achievement/Trophy for doing this, and the game keeps a tally of "Pants Exploded." On the GNR program "The Adventures of Herbert Dashwood," Argyle calls this technique "the ol' Shady Sands Shuffle."
That same game, however, follows the same "essential NPC" rules as Oblivion — plot-critical characters do not die (and, since this is a Fallout title, they must and do have protection from possible dismemberment). Children cannot be targeted at all; they have no collision detection for attacks, melee or ranged, and they can't be targeted in VATS mode.
In Fallout: New Vegas however, while children remain invulnerable almost other NPC can be killed no matter how important they are. The exceptions are General Oliver if you're an NCR courier and Legate Lanius if you're a Legion courier. The two are only seen for a few seconds at the end of the game and killing them would render the game unwinnable. If you play as House, his Securitron is also invulnerable during the endgame quest.
Additionally, the bodies of Victor and Yes Man can be killed, but they'll just show up later because they're AIs that can jump from one body to the next.
In Fable, your weapons are taken away from you in Bowerstone, which is merely cosmetic because you can't beat anyone to death there, either (naturally, that's where the game puts all the children). Unfortunately for the people of Bowerstone, the game is only nerfing your damage, and if you can bring in and subsequently protect a mercenary or two, you can murder your way across town all you like!
In addition, you can always pick a fight with the guards and navigate the town in a way that gets the townsfolk killed by friendly fire.
And also, one of the simplest mods to make for Fable is one that lets you keep your weapons in Bowerstone, with predictable results.
Fable II has so many Lord British-type NPCs, it's not even amusing. Notably, you cannot attack when anywhere near Theresa, and if you find a way to snipe her from range with Skill or Will, she will tell you that your weapons have no effect on her.
Median XL is a Diablo 2 mod that features invulnerable trap-like monsters that kill you instantly when you get near, but don't move. This being Diablo, you can imagine how well that went.
Not only that, but players figured out ways to kill monsters that are immune to all elements, monsters that cannot even be targeted (tip: damage reflection) and even monsters that cannot be targeted and are immune to all damage and do not have any attack that would trigger damage reflection. There are even four different ways to do so: resurrect monsters with "burn" damage which bypasses resistances and attacks hit points directly; find the item that summons instakill reanimates on your side when you kill enemies; find the item that summons a certain boss whose death animation is coded to autokill nearby monsters on your side and let it "expire"; or use a bugged passive in combination with Open Wounds to cause the monster to take physical damage despite being immune to it.
In Torneko: The Last Hope: there are priests in Toro Ruins, and they are treated like monsters, except you can't attack them directly. But you can shoot arrows, magic thunders, or even turn them into items. When this occurs, though, a message appears and says: "Divine retribution!" and a giant lightning bolt drops your HP to 1. Oddly, this is recorded on your adventure log.
Kingdom of Loathing had a rumour/legend that the Hermit NPC was killable, via some sort of trickery. As a browser-based not-so-multiplayer role-playing game, he has no stats and no way to be encountered as an enemy. He's said to 'drop' most non-obtainable (or even nonexistent) items in the game.
The final boss of Kingdom of Loathing is the Naughty Sorceress, who is supposed to be killable only with a certain item in the inventory. Two players, Cobain Dougans and DarthDud, managed to beat her without the special item and were rewarded with custom Golden Sausage and Silver Sausage items. The description on the Golden Sausage reads, 'Congratulations on your surprising victory, and darn you for forcing us to come up with a way to keep that from ever happening again.'
Cobain Dougans has also managed to defeat the Guy Made of Bees, who is also supposed to be unkillable without the use of a certain (different) inventory item.
The Sorceress has been beaten a third time by MimiRiceCat. No Bronze Sausage yet, but the Sorceress has been upgraded again. It is now no longer even theoretically possible to kill her without the necessary item—you instantly win if you have it, and instantly lose if you don't.
It should be noted that while the others had to get crafty about it, Cobain was one of the highest leveled players in the game, and accomplished this through SHEER FORCE. In a game where the average player ascends by level 15, and there are trophies for those that get to 30, and most clan-quest people hover around 40, he was 116 at the time. Note that leveling up requires you to get a specific base stat (which one depends on your class) up to (current level^2)+ 4, and to get the nth point in a stat requires n^2 total EXP in that stat. Cobain Dougans' screen shot shows◊ that he had a base Mysticality stat of 13354, buffed up to 48476. This means he had over 178.3 million EXP in his Mysticality stat, when the only enemies at the time with a base yield of more than 40 EXP were for seasonal special events. Determinator indeed.
It happened again. Even though you instantly lose if you don't have the item you need, a glitch in a newly implemented item gave a player a chance to act, upon which they made use of a different newly implemented item to reset her stats to manageable levels. This would serve as a testament to the ongoing creativity of players, except that it apparently managed to happen entirely by accident this time.
In the last level of NetHack, a player will meet the Riders, three immortal beings who, when killed, will always rise from the dead again. There are only two known ways to dispose of them permanently: kill them, and fill the level entirely with monsters so they have nowhere to reappear, or turn them into green slime. Neither is considered a bug (in fact, there's a special message for the former), although most players consider them much more of a hassle than simply finishing the game.
Amusingly, it's also possible to level-drain two of the three until they become weak enough to enslave with Charm Monster. Now you're playing with power!
Some players like to take this to the next level, and kill entire species. This is easy for most monsters, as almost any monster that is created 120 timesnote 3 for the Erinyes, 9 for the Naz'gul, and never for the Keystone Kops is considered extinct by the game, and will never be randomly generated again. You'd think this wouldn't apply to unique monsters— by definition, they are created only once, yes? But in fact, a monster that is brought back somehow counts as a new monster for extinction purposes. Stoning and unstoning a monster counts. For monsters that resist stoning, a wand of undead turning will resurrect the monster. For those few stone-resistant monsters that leave no corpse, an Amulet of Life Saving will also increase the extinction counternote gathering 120 of an already-rare item for each unique monster is an exercise in madness, but can be done— by invoking Wizard Mode, if nothing else. But there is one monster, Juiblex, who is unique, stone-resistant, leaves no corpse, and will never pick up an amulet (because he has no neck to wear it on)... but even he can be made extinct, if you let him engulf you and drop the amulet in his belly, then cast slow or speed monster on him to (somehow) make him wear it; a technique which was discovered only in June 2012, when the last update to the game was December 2003. Extinctionists take their job seriously.
In earlier versions, it was possible to tin Death, removing him from the game permanently. Eating the contents of the tin was instantly fatal.
NetHack also gives us the Mail Daemon, a creature that delivers messages from other players watching you online. It appears and disappears within the span of one turn and, therefore, can't normally be interacted with at all, much less killed. So, naturally, sufficiently Crazy-Prepared players have found a way to kill it. To do this, the player needs to wish for a statue of a mail daemon, cast stone-to-flesh on it and to kill the resulting daemon in one turn before it disappears with a cry of "I'm late!"
The first Valkyrie Profile game has Brahms, Lord of Vampires. You are not supposed to beat him. If you do through luck, Level Grinding, or cheating, the game just deposits you back into the Overworld without ever mentioning it again, and the game pretends that you had chosen the "don't fight him" option. He appears again in the Bonus Dungeon, but by that time you are expected to be strong enough to match him.
In the US version of Final Fantasy IV, Final Fantasy II, Kain is intended to be invincible when Cecil fights him over a crystal after Kain's under the influence of Golbez's mind control. However, though Kain has 60,000 HP, a ridiculous number of health points for an enemy so early in the game, one can use a combination of level grinding Cecil to 40+ as well as unequipping all of his armor except for his shield to make Kain's "jump" attack miss virtually every time, allowing one to slowly drain Kain's HP and heal Cecil before Kain can inflict enough damage to win. Of course, though Kain turns purple and fades away, as if a normal enemy has been defeated, the scene immediately afterward goes on as if Cecil had lost the fight, not acknowledging anything the player accomplished in battle.
In the original NES version of Final Fantasy III, you have what is supposed to be a Hopeless Boss Fight against Bahamut early in the game. He has 65535 Hit Points (an absurd number for that game), and fully heals every single round, but it is still possible to kill him. If you do, not only do you get no reward, but you get punished — it becomes impossible to get him as a summon later in the game.
Note that 65535 is obviously 2^16-1, and so is likely the maximum possible health the technical limits would allow the devs to give him.
Also note that PC damage is capped at 9999, and the party limit is 4. At that point, it's quite impossible to deal more than 39996 damage in a round, then he full-heals. (In reality, that 39996 is more akin to, say, 400.)
In Final Fantasy VI, you have the Guardian, which is supposed to be invincible and act as Border Patrol, blocking entry into Vector at a couple points in the game. It can be defeated due to the Echo Screen/Joker Doom glitch (which allows you to instantly win any battle.) As expected with this trope, the game acts as if you lose. The glitch was fixed in the GBA port.
Giygas is supposed to be killable through one means and one means only: Paula's "Pray" command. While he does have a "real" HP value, calculational trickery otherwise keeps him alive. However, in the Game Boy AdvanceCompilation Re-release, using a Viper on Giygas can poison him... and the poison will eventually kill him. On the other hand, since the battle wasn't ended the scripted way, glitchery ensues and you end up in a world preconfigured for the game's ending... an ending which won't happen now.
Master Belch is immune to damage of any kind until you use the Fly Honey on him. However, the Hungry HP-Sucker doesn't count as damage, because it's a draining effect. It's possible to kill him the long way by exploiting his vulnerability to status effects and drawing his hit points out the long way.
In MOTHER 3, the only way to kill the Bonus Boss Pig King Statue is supposed to be with either a lucky shot of PK Flash, or use the New Year's Eve Bomb. He has 100 million HP otherwise, and regularly hits you for more HP than you probably have. It's just not possible to beat him conventionally without cheat codes, if only because, at the end of the day, long after you've run out of PP and healing items, he still has 99,000,000 HP left to go. It's a numbers game and his are just far, far, far superior to yours. That hasn't stopped people from trying, though. Problem is, for the people strong enough to go the distance, it turns out he has another dirty trick: freezing the game. He hadn't even worn down all 99,999,999 HP — according to a comment, it was somewhere around 2 million.
The Ultimate Chimera avoids this thanks to the fact that the game doesn't even give the player the opportunity to fight it. If he catches you, it's an instant Game Over.
Secret of Mana has Charon, the guide who takes you to the Moon Palace, who is actually just a common enemy called a Robin Foot locked into a position where he cannot attack the player. He can't be hit with weapons, but magic can kill poor Charon. Additionally, the final boss is supposed to only be damaged under very specific circumstances ( having the hero hold the Mana Sword and having both of his sidekicks cast Mana Magic on him), but his defenses are not impenetrable to normal weapons when charged to sufficient levels.
Disgaea has another example of this. NPCs are everywhere across the castle - Running the hospital, the shops, even the portal that takes you to the storyline maps. Now, you can't attack them in the castle.... or can you? (Here's a hint: You can.) All you have to do is call an assembly. If one of the NPCs shows up, you can happily beat the tar out of them. Then when you get back to the castle, you'll find a little gravestone where they were. Wait a minute.... you just killed the demon running the Hospital... how are you supposed to heal? Just save, reset, and they'll be back in their spots again.
The series also features a few Hopeless Boss Fights that you must lose to advance the plot (Etna and Laharl in Disgaea 2, for instance). These enemies are never truly invincible, just way more powerful than your party is expected to be. This being Disgaea, you can stop and level grind for ages until you are capable of winning. Typically results in a Nonstandard Game Over. Since seeing all the different endings is a frequent goal of players, plenty of people actually do this.
In Makai Kingdom, at one point in the main story you face off against Zetta's nemesis Alexander, who is level 1000, which you'll have no chance to beat if you''re only leveled appropriately for the previous fights. However, Salome joins at level 1200 as a neutral party and almost certainly kicks his ass. However, with some dedicated power leveling/grinding/New Game+, you can either beat them both, or assist Alex in winning over Salome and then beat him. This triggers a new fight where Zetta wants revenge on Alex for killing Salome (yes, even if his team killed her). Winning leads to a Nonstandard Game Over and a New Game+ where Alex is now part of your army.
Most of the townsfolk that aren't important in some way or merchants can be killed in Final Fantasy Adventure if you have the patience to do so.
It's actually possible to kill Fargus in Fire Emblem: The Blazing Sword. You're warned not to do so because it's a Hopeless Boss Fight otherwise, but people have managed to defeat him legitimately. Alas, it's not a good idea to do this because this causes a game over just for attacking him.
Almost the exact same instance occurs in Radiant Dawn, possibly making it a Shout-Out - the Big Bad appears in chapter 4-3 amongst his hordes of Mooks. He's supposed to flee once a certain ally appears during the battle, but it's possible to get to him and take him out first. As above, he makes a pompous exit and returns later in the story no worse for the wear.
Evil Islands has many monsters that were designed as unkillable by giving them tons of HP and rapid regeneration. However, with the introduction of easy mode in a patch, most of them became technically possible to kill, even though the process was long, difficult, and involved a share of luck. Enthusiasts posted a detailed guide to killing every single creature outside of towns, except two dragons that are too tough to kill even this way and a frog in the tutorial, which only survives because the player has no ranged weapons at this point. This slaughter, nicknamed "Project Genocide", completely breaks the game scripts, making quests play in the wrong order, NPCs making references to future events, and corpses and empty spaces participating in conversations.
Chrono Trigger: during the first encounter with the Big Bad, its stats are actually inflated to ensure you lose. However, with sufficient grind or on New Game+, you can kill it and witness a truly odd alternate ending.
In MARDEK Chapter 3, when you go through the Warport you get barraged with security questions from thinly disguised TSA guys. If you give the wrong answer * "Do you have your ticket?" "No, I burned it, like I'm gonna burn you!", you get to fight a Security Demon, which is immune to your best attacks, has 50000 health plus regen, all of its attacks are one hit kills, and can't be summoned by the time you're strong enough to take something like that in a fair fight* After five uses of the Warport, you get a Gold Warport Pass, which lets you bypass the security questions. That tends to happen before you reach any of the best grinding grounds. However, by having the right party members, spamming the right skills and being crazy lucky, it's possible to kill it. It gives massive amounts of XP, and the security guy grants you the Gold Warport Pass early since obviously they can't stop you even if you are a terrorist.
Just as statting the Lady of Pain is discouraged for this very reason, her appearance in Planescape: Torment also averts the trope by never even letting her appear outside of FMV cutscenes, where she promptly either teleports the Nameless One to a Pocket Dimension maze or just straight up eviscerates him.
Toady's stated design intent to make it possible to annihilate all life on a generated world, ushering in the Age of Emptiness.
Most creatures can be killed by crushing them under a drawbridge. Large enough creatures however will instead crush the drawbridge. Still these creatures can be killed using drawbridges - by putting them in a cage first.
Alter Aeon seems to enjoy tormenting players with this: in the starting city, there is an NPC called "The Captain of the Guard". The message for the area explicitly tells the player (paraphrased) "This is the Captain of the Guard. He cannot be killed. No matter how many players you bring, you will not kill him". Naturally, many see this as a challenge.
In Zelda II: The Adventure of Link, Bubbles are supposed to be invulnerable, but due to programming limitation are actually set to take 255 hits before being defeated. If you have enough patience it can be done for a measly 50 experience.
In The Legend of Zelda: Link's Awakening, chicke... er, Cuccos and dogs can't be killed with the sword, attacking the former enough causes a whole flock of Cuccos to start swarming Link (as is the norm in Zelda games), and attacking the latter triggers a counterattack. Use the Fire Rod or Magic Powder, however, and not only can you kill them, but an active Cucco swarm will stop. (There's loyalty for you!)
Immortal Defense has a boss at the end of the second campaign who's supposed to be a Hopeless Boss Fight, but players who are just that good have gotten him. The developer didn't think anyone would be able to do it and the game assumes the normal events happened where the boss wins and continues from there. The rest of the game, heavily psychological, goes on to make near-record amounts of no sense.
In Age of Mythology: The Titans expansion, one level has you scrambling to revive a giant Guardian statue in Egypt, because an equally giant Titan (Cerberus) is coming who is designed such that killing him should only be realistically possible using this guardian, due to his massive damage and hp. However, using vast number of priests, who gain damage bonuses towards creatures of Myth, it is quite possible to kill him without awakening the guardian, though he does tend to kill a lot of them before he goes down.
You can kill Gargarensis forever, preventing him from resurrecting, by building a house upon his corpse.
One of the missions of Necropolis scenario in Heroes of Might and Magic V: Tribes of the East features a fortified Orc town with a huge army* and "huge" means "at least several hundred cyclops, and according number of other creatures". This army never leaves the town, and the game specifically warns you that you don't have to and shouldn't attack it. However, it is possible to destroy it, by prudent use of Dark magic (specifically, Berserk and Puppet Master spells).
Cap'n Hector from the Escape Velocity games was an NPC who would remind players to register (pay for) the shareware game. After the 30 day trial period ended, the Cap'n would start warping in to attack your starship every chance she got, usually resulting in a very quick death. However, while Hector was invincible to conventional attack, it was possible to kill or disable her with splash damage in the original game.
Nothing is ever unbeatable in a Nippon Ichi game. With some clever tactics, decent equipment and/or lots of grinding, anything can be defeated, and doing so may even net you special rewards in the form of rare gear, named characters and special cutscenes or endings. Be aware that defeating some enemies you were supposed to lose to may result in Bad Endings.
Sandbox Game Terraria allows players to kill the neutral/friendly NPC characters in their village under some circumstances. The Clothier drops his red hat when you take him out, and the Guide can be directly killed if you equip the Guide Voodoo Doll and attack him, or if you dunk the Voodoo Doll in the lava in the Netherworld. You HAVE to kill him by the latter method to spawn a certain boss that must be beaten to progress to Hard Mode. Also, the Dungeon Guardian NPC, meant to protect the Dungeon from earlier acess than intended, had 9999 hp, 9999 defense and 9999 damage, aside from being very fast. Yet, many players were able to kill it, using fast shooting weapons (that would deal 1 damage per hit) and reversing gravity in order to be fast enough not to be hit.
Nothing in Kerbal Space Program is supposed to be destructible, other than your ships and crew, and there isn't much else in the game besides planets. So, naturally, players found physics glitches that allowed them to destroy planets.
Danger Ranger, the obnoxious jet-engine on skis that pesters you incessantly in Slamscape can be killed with the crushers found in Repsychler or Viva Los Vangrantes.
In the Freespace series, there are some missions you are supposed to fail, and some ships that are supposed to survive and escape to fight another day. Naturally, some players attempt to Sequence Break the game by killing these ships. There is an "Invulnerability" flag for each ship that can be triggered by a mission designer to prevent this, but many mission creators consider using it to be cheap and sloppy. Thus leading to this trope when they try to engineer a situation in which it seems logically impossible to defeat an enemy, yet some player manages it anyway and breaks the mission.
The original Metal Gear Solid included a fight with psychic Psycho Mantis, who would dodge all your attacks because he could read your mind. In order to hit him, you were supposed to shut down his mind reading ability by plugging your controller into controller port 2, but there are brief moments where you can make a hit connect even without doing this, letting you (slowly) win the fight with the controller still in port 1.
Non-video game examples:
In Star Trek II, the Kobayashi Maru scenario (the original trope namer for the Unwinnable Training Simulation) is designed as a Hopeless Boss Fight for cadets, more as a psychiatric evaluation than a tactical scenario. It involves trying to rescue a stranded ship in Klingon space (later in-universe iterations change the "enemy" race as Society Marches On; a similar scenario depicted in Star Trek: Voyager uses Romulans). Captain Kirk relates the story that, as a cadet, he hacked the program to make it winnable, and received a commendation for original thinking in the process rather than being reprimanded.
A novel called Kobayashi Maru had Enterprise crew members tell stories of how they dealt with the scenario. Kirk's story revealed that he actually went up against the no-win scenario multiple times in fruitless attempts to beat it. He eventually decided that since the computer was cheating to make the scenarios unbeatable, it was only fair for him to cheat back. So he reprogrammed the scenario to make his opponents recognize him as The Dreaded, allowing him to beat his opponents merely through the reputation he hoped to build in the future. The sheer audacity of the move was what allowed him to get away with it.
In the 2009 pre-boot, he does the same, but based on his attitude in the changed timeline he gets in trouble for it.
The Expanded Universe revealed that now cadets are encouraged to find solutions like these.
One of the "New Frontier" novels revealed that Captain Calhoun's solution was more similar to this trope than the others- instead of trying to rescue the Kobiyashi Maru, he blew it up and ran like hell, reasoning that either a) they were spies, as that's the only reason a civilian ship would be in the neutral zone, or b) even if they weren't it wasn't possible to rescue them, and he was sparing them from being sent to prison camps.
The Matrix has Agents, entities the Machines use within the Matrix to hunt down runners and rebels. They possess the bodies of normal humans, making them impossible to kill permanently (as they'll just leave the dead body and possess a new one). Even killing them temporarily is all but impossible, as they're very good at dodging bullets. But, at the end of the first movie, when Neo embraces his identity as the cyber-messiah, he actually does kill Agent Smith by directly screwing with Smith's code.
In the sequels, Agent Smith actually returns from death—rather than submitting to deletion, Smith transforms himself into a computer virus. By the end of the third movie, he's infected every program and human in the Matrix. This makes Smith both impervious to deletion attempts from Machines outside the Matrix, and able to defeat Neo within the Matrix. Smith is finally killed for good when Smith possesses Neo, which makes him vulnerable to a kill-code the Machines inject via Neo's body.
In-canon example: in Caprica, this is probably why "New Cap City" players keep on trying to kill Zoe and Tamara, despite the inevitable results.
An old adage in tabletop RPGs, and similar to this trope, is "If you stat it, they will kill it." This is probably a take-off of the Predator quote, "If it bleeds, we can kill it."
This may be why various RPGs avoid statting certain characters. For instance, Spirit Of The Century gives only a few ballpark skills for Dr. Methusala, but only as a last resort. GMs are encouraged to treat him as an event, rather than a character.
One possible risk with this approach is that in tabletop games players may take it more personally than in other media. After all, what looks like an "obvious" reason to make an NPC invulnerable to one person can easily come across as simple game master favoritism to another — and unlike most more distant creators, the GM is generally sitting right at the table with the players (and can at least in principle always overrule any offending "official" stats anyway, so even those only go so far as an excuse).
Similarly, the tongue-in-cheek First Law of Munchkinism: "Any finite number can be reduced to zero."
Averted in the Babylon 5 RPG. The rules of combat for Vorlons are essentially "You Lose."
The Original Edition Gods, Demigods & Heroes and the 1st Edition Deities & Demigods Cyclopedia gave the gods stats. This led a lot of players to treat them just like really tough monsters and try to kill them. Later editions generally avoided statting the gods themselves, and if they needed to make a personal appearance, would send an avatar with a fraction of their powers (but still really tough).
The 3rd ed book Deities and Demigods contained almost exclusively stats and info on most gods in the game, including the Faerunian pantheon and the Greek, Norse and Egyptian pantheons. A majority of them are grossly underpowered and badly built, ripe for having player parties kill them and steal their divine powers.
The Planescape entity known as the Lady of Pain has no stats for this very reason. As the inscrutable ruler of Sigil who can flay an opponent with her shadow and keeps major gods at bay, she's not supposed to be taken down by a bunch of munchkins. The fans wouldn't have it any other way, to the degree that when the 3rd edition Planar Handbook statted her Character Alignment and nothing else it was met with disapproval.
Writers for the Ravenloft setting split the difference with Gwydion, an Eldritch Abomination: they wrote up stats for his tentacles, but never give the heroes a chance to confront his actual, unstatted body.
In a variation, the Tarrasque is designed to be the most powerful monster in the game - it's huge, it's strong, and it regenerates any damage quickly. Theoretically, you could beat it, but the rules are stacked against you, so players gave always been trying to come up with creative ways to fight it. One thread on rpg.net described how the rules make it entirely possible to have a city capture the Tarrasque and use it as a neverending supply of food, since it'll always regrow what it loses.
Pathfinder clarified that their Tarrasque was unkillable, period, even with wishes and instant death spells. It will always regen the damage eventually. Of course, making it someone else's problem is not out of the question.
In contrast, all major characters in Eberron are quite killable, by design. The Lord of Blades, a major character in the setting, is only level 12. Vol, a being worshipped as a deity by thousands, is only level 16 with some templates. Of course, a level 12 3.5 character is still ridiculously strong by any reasonable standard.
Faerun's major players are also mostly statted out. Though they're generally higher-level than Eberron's, they're not designed very well, being hastily converted from their AD&D stats. Evil adventuring parties tend to treat Faerun like a big theme park full of low-hanging branches of XP.
Achaekek the Mantis God has an ability which stops any mortal dealing him lethal damage... but it doesn't work in an anti-magic field.
This is also part of the reason why most (if not all) of the monsters that had been Good-aligned in previous editions got reset to "Unaligned" at best in 4th edition: A creature doesn't need stats if you're not going to fight it, and why would your characters fight Good creatures (seeing as no DM in his/her right mind would allow an openly evil party).
The crew at Pinnacle would recite the adage about statting things in the first edition of the books. They would refuse to stat certain characters in order to railroad people into sticking with the metaplot. Seems to have been reversed in the Reloaded release, though.
The character Stone has no stats for this exact reason. In Deadlands Reloaded, he has stats, but takes no damage from anything other than a gun he fires himself, or the bullets that killed him the first time. Neither of which is going to happen.
The module "Harlequin" refuses to stat the title elf, and explicitly calls this out as the reason. (Great Dragons, meanwhile, avoid the Postulate not by not having stats, but by having a mechanic in their stats that allows them to say The Battle Didn't Count, to appear dead but actually survive, and then destroy the PC party via manipulation from a place of complete safety.)
Harlequin finally got statted in Steet Legends, which is full of stats of famous Runners. If you though Harlequin was tough, Gwynpaine is much worse. And unless you perform massive overkill, Harlequin will appear later and state he had a nap and may let you off seeing how you amused him.
As of the fifth edition, all dragons now have Edge and great dragons no longer have stats in the rulebook. There are guidelines for making them, however, but given how fierce a regular dragon is, the concept of fighting one, let alone getting away with it, is a pipe dream for most runners.
The third edition of the book made sure to give all vampires of third generation and below at least one Plot Device power to keep the players from just killing them willy nilly. Which is amusingly inverted in the Time of Judgement, which has a couple of scenarios that go out of their way to give players ways to kill (or help kill) 3rd generation vampires (and Caine!) willy nilly.
The "you lose" rule is also used in the LotR RPG to describe fighting Sauron.
The older Middle Earth Role Playing gave statistics for Sauron in the Necromancer of Dol Guldur supplement. He's immensely powerful at 240th level (for comparison, Nazgul are 20th level in the original book or 40th in the supplements and Gandalf et al are 60th level), but since Middle Earth Role Playing has open ended rolls and instant-death criticals, even the lowliest hero could get lucky and kill him with a lucky combination of 96-100 attack rolls and a decent critical hit, assuming they could get into combat with him in the first place.
Pendragon's combat system works on a basic d20 to hit. You have a skill, roll under it and you hit and between you and your opponent, the highest hit wins. Roll your skill exactly and you critical; A critical is better than a hit so it's possible to defeat someone of higher skill in a pass of arms with a critical even if their skill is higher and they roll a higher hit. When your skill reaches 21 you critical on both 20 and 1; 22 is 20, 1, and 2. Lancelot has a 40 skill with all weapons; he always hits and he always crits. Yet it is possible to beat him in a pass at arms and even in an entire fight (as happens once in the entire Morte d'Arthur). If you always crit higher than Lancelot's crit, you can damage him through his usually-amazing armor, his usually-present protection from God, and his slightly-superhuman hit points until he falls. Galahad is another story. Under Galahad, it reads, "Galahad wins."
The first of the combat abilities of Cthulhu reads: "Each round 1D3 investigators are scooped up in Cthulhu's flabby claws to die hideously". (Through Memetic Mutation, or perhaps from an earlier edition, this often gets quoted as "Cthulhu devours 1d6 investigators per round" or the like.)
Early editions had Cthulhu eating 1d4 PLAYERS a round.
Just seeing Cthulhu forces a character to lose 1d100 SAN (sanity). The maximum SAN is 100. Do the math.
Though the thing that makes him this trope is what happens if you do kill Cthulhu. He reforms a few minutes later, fully recovered. Nuke him and he comes back radioactive.
In the Cthulhu-based boardgame Arkham Horror, it is possible to have to fight and win fights against the Eldritch Abomination the players picked at the start of the game, including Cthulhu himself. The exception is Azathoth. While other bosses may have very nasty and complicated attacks listed on their sheet, Azathoth's attack description is simply this: "The end is here! Azathoth destroys the world."
Rifts being Rifts, when it added "Pantheons of the Megaverse", the first thing many players tried to do was take on Odin, Ra, Marduk, or whoever was handy. But the book also described beings far more powerful with lines as succinct as "tick them off and it's time to roll up new characters." No stats or names were provided, but presumably the writers were referring to capital-G God and distinguishing Him from those wannabes on Olympus. (It also suggested that Game Masters didn't include Him in the campaign, since constant divine intervention makes the game kind of boring.)
Full character stats for such in-universe celebrities as Erin Tarn and Emperor Prosek are presumably included in the game just so players can take a stab at offing them; named characters without stats are actually a minority.
7th Sea has one of its most infamous villains, Captain Reis, on the cover. He has stats. He's also nearly impossible to hit... with a sword. However, he doesn't wear armor and you can't block bullets. Shoot him and throw in all your drama dice and you can put a hole in his skull.
Celestine the living saint has an ability which means that she can never be killed — rather, if you do, she will get back up. The only way to stop this is to use the special character known as Old Zogwort, who has a 50/50 chance of changing her, special rules and all, into a small creature called a squig.
Sadly, as of the 2014 Ork Codex, Old Zogwort is no longer available.
You can also remove her special rule that stops her running away, and then make a Sweeping Advance when she does run - the rules for Sweeping Advance state that nothing can save you from dying.
Notably, this is NOT possible by trying to turn her into a Chaos Spawn, because it kills her and summons the spawn rather than altering her profile.
Depending on how you interpret the rules, this may or may not be possible using a Vortex Grenade. It prevents use of saving throws or "Any clever way you can think of of staying alive." The debate is over whether she is staying alive, or dying and getting back up.
Yet another possibility is a very lucky shot with a Shokk Attack Gun. Managing to a) hit the enemy and b) rolling a double-six results in deamonic slime raining from the sky onto the target with the rules explicitely stating that anything hit by THAT is automatically removed from play.
In the New World of Darkness, it's directly stated that the Leviathan can't be killed or even stopped in any meaningful sense — the stats provided are solely for escaping it, or hurting (more like "annoying") it enough to drive it away temporarily.
Promethean: The Created notes that there is no significant way that any creature of this world can hope to damage an Arch-Qashmal. If it can even be bothered to retaliate, it can wipe out the players with a thought. The book that says this does, however, add that intrepid players can destroy it indirectly; all qashmallim are created to fulfill a specific goal, so if they can render its goal impossible, the Arch-Qashmal will cease to exist.
In the webcomic Kid Radd (which plays with and lampshades a number of videogame tropes) the final boss of the game Mofo was supposed to be only vulnerable to a special ability none of the characters had. It turned out that he just had a lot of hit points and regenerated them each turn, which allowed the team to defeat him.
In Homestuck, the leader of criminal gang The Felt is named Lord English, after a particular spin put on a cue ball in pool. Creator Andrew Hussie hadn't even heard about Lord British until later and, when he was informed about similarity between the names, decided to make Lord English apply to the Lord British Postulate, mentioning in a character's narration that he can only be killed by exploiting "numerous bugs and glitches in spacetime."
This particular Retcon surprisingly fits very well within the Time Travel-related powers that the rest of The Felt possess.
Unfortunately, after too much of this, they became hostile and actually did begin attacking him; how he managed to fight off and eventually kill the entire species is a mystery not explored by the comic.
Karkat even mentions that the Angels didn't drop any loot so it was obvious they weren't supposed to be fought.
This is Vriska's attitude towards Bec Noir the Omnicidal Maniac created when Becqueral prototyped himself, inadvertently giving the Kids version of Jack Noir (who had already gone insane) a massive power boost. Specifically she wants to be the one to personally kill him (despite how ungodly powerful he is) to "win" the game. Naturally it turns out that she massively underestimated his abilities and Terezi is forced to kill Vriska to keep her from leading Bec Noir to their hiding spot (which, in an alternate timeline, caused the deaths of all the Trolls).
Additionally, their guidelines on how to kill RealityWarpers involves a bit of this. They might be able to erase you from existence itself with a thought, but they're still limited by human intelligence - the trick is to come up with ways of killing them that they won't see coming.
In "All Jedi or No Jedi", he mentions having put a cameo of Darth Vader into a Star Wars roleplaying campaign just to add a bit of flavour, only for it to instantly derail as his players become obsessed with trying to kill Vader. Spoony refuses to let them and tries to nudge them into realizing it without stating it, but they still pursue Vader and end up being killed when he escapes and they're surrounded by enemies.
In "Because He's There" and "If You Stat It, They Will Kill It" he criticizes the use of the D20 system for games involving entities like Cthulhu, because if the designers give something stats, inevitably the party will try to kill them. Even though they are such strong stats that it is almost impossible to actually defeat Cthulhu, and knowing he will just regenerate even if they somehow do, players will still always try because of this trope.
Similarly, "Thou Shalt Not Fuck with the Lady of Pain" discusses a campaign where a player decided to try and show up a being who's more powerful than gods; thanks to Spoony's ability to think on his feet, it didn't happen.
Spoony spoofed this with the original Lord British himself, Richard Gariott! As his "Avatar" self, he tries to attack Lord British, who doesn't even notice he's being attacked.