Kudos (Greek!) to Hercules for thinking outside the ampitheatre on this one [...] but it seems to me there was another way out. If he had just kept chopping, eventually the creature would have had a hundred thousand heads, making it look something like venomous reptilian broccoli. Then it would have tipped over and been no threat to anyone. People could come up and laugh at it, it would have been a great tourist attraction.
This is where when you defeat an enemy, one or two more show up in its place, and so on until you perform some kind of specific attack to kill it for good. Named for the Hydra from Greek mythology, which had nine heads that grew two more heads whenever one was chopped off; the only way to destroy it was to cut off the heads and cauterize the stumps with fire before the head was able to grow back and multiply. Compare to Asteroids Monsters, in which a monster when destroyed divides into several smaller versions of itself and you have to keep killing them until they are dead for good. Expect a lot of decapitation, even though this is probably the worst possible strategy to use.
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The Monster Society Of Evil has a literal Hydra created by Mister Mind, which when it loses a head grows it back with the head of another animal. Captain Marvel causes its heads to fight over meat, killing it.
Madrox the Multiple Man of X-Factor has the power of Me's a Crowd, but it usually works like the Hydra Problem from his enemies' point of view because he involuntarily creates a copy of himself when struck. Punch him, and suddenly he has backup. The answer to beating him is a One-Hit Kill (Pretty Little Headshots don't create copies), restraining him so that he can't receive a sufficient blow (punching a wall would suffice, so you'd better tie him up tight), or employing a method that doesn't involve direct force (poison, for example).
This concept is the origin behind the name of the terrorist organization HYDRA in Marvel Comics, though in practice it's essentially a boast about their never ending Redshirt Army. Their motto is "Cut off one head, two more take its place". In Captain America: The First Avenger, this sets up one of Colonel Phillips' (played by Tommy Lee Jones) CMOA when, after shooting a HYDRA mook who just shouted the line, quips "Let's go find two more." It also sets up the many times that an Anti-Hero decided to see whether the creed was literal at an individual level: with all the superheroes and supervillains running around, it would after all be easy to make an army of mooks with a twisted healing factor. The surviving mooks are often quite dismayed to realize that it's not.
The Superman villain Riot has a similar power to Multiple Man. Defeating him requires indirect methods like cutting off his air supply or catching him in a net.
Films — Animated
There is also the Sorcerer's Apprentice segment from Fantasia. When Mickey chops the broom into pieces, each splinter becomes a new broom. Only the Wizard's magic can undo all the damage.
The Disney adaptation of Hercules had Herc face this problem when fighting the iconic multi-headed serpent. In desperation, he starts wildly chopping at the beast, leaving him with a bigger problem than what he started with. However, he soon triumphs by causing the chasm they're in to cave in, resulting in the Hydra being crushed under a rock slide.
Films — Live-Action
In the first Hellboy film Sammael is a creature that, when destroyed, gives life to two of its previously laid eggs. The only way to stop Sammael is to wipe out all of its bodies and eggs at once.
While appearing in the movie of Percy Jackson and The Lightning Thief in the books the Hydra appears instead in Percy Jackson and the Sea of Monsters. Here, Clarisse kills it by blowing it up using a gunboat.
In Prometheus, Fifield attempts to cut off the alien snakes head when it breaks Millburn's arm but not only does it spray acid blood over Fifield's face for his efforts, the head instantly grows back.
In The Andalite Chronicles, Visser Three unleashes mortrons, creatures that can regenerate into separate beings when sliced apart. Elfangor resorts to knocking them out. Loren strangles one to death, and uses a softball bat to cave one's skull in.
The Angel episode "Waiting in the Wings" had mooks who would respawn into two whenever they were killed. Angel had to fight past them and defeat the mage controlling them to make them disappear. Fortunately each time a new lackey is created, the mage's power weakens as he has to keep control of an increasing number of mooks.
On Charmed Swarm Demons are replaced by two more swarm demons when they're killed. The sisters have to destroy the lead Swarm Demon, the demon from which all others come from, in order to kill them all. And in another episode an evil witch unleashes her snake familiar onto the sisters and when the snake is chopped into two pieces, the pieces both grow into a new snake. The witch is killed and then so are the snakes.
Mythology and Religion
Trope Namer comes from the Lernaean Hydra from Classical Mythology. Whenever a head was broken, two more grew in its place. To defeat it, the hero Heracles had his friend Iolaus burn the stumps before the head grew back until he smashes off all the beasts heads with his club.
The Hydra also had one immortal head, which they had to bury under a rock.
Parody hero Samurai Cat fought Cerberus, who it turned out had the same power. Samurai Cat just kept chopping off heads until Cerberus had so many, 1) they were all too small to bite effectively and 2) they weighed so much Cerberus couldn't walk around anymore.
In the Hindu Mythology Durga once found herself facing an enemy whose power was a regeneration-based Me's a Crowd - from each drop of blood spilled, a clone would pop up into existence. After a moment of futile fighting, she transformed into her Kali aspect, which quickly solved the problem by catching her enemy and eating him whole.
More Hindu mythology: The legend behind the Thuggee cult was about regenerating demons too, therefore demanding strangulation (a bloodless killing method).
The Bible [Jesus; Matthew 12:43-45] once said that if you defeat a demon/unclean spirit with your own power that isn't God's, it would return with seven more of its kind.
Judges Guild module Dark Tower. It had an area with a mirror hanging on the wall that was guarded by a skeleton. If an attack against the skeleton didn't do exactly 8 Hit Points of damage, it would split into two skeletons with the same property. The only way to defeat the skeletons (other than by doing damage) was to destroy the mirror.
D&D's Monster Manual also lists the Hydra among the monsters that the players can fight. However, only the Laernian variety of the hydra exhibits the Hydra Problem; the (normal) hydra and the firebreathing pyrohydra both get weaker as you lop their heads off, not stronger.
Magic: The Gathering has a number of creature cards that implement this trope as mechanics. Usually, if the creature survives taking damage, it gets stronger afterwards.
Hydra Slayer is a quirky little Roguelikegame wholly concerned with how to resolve the Hydra Problem on a case-by-case basis. Your success is determined by being able to tell which weapons/powers will remove heads, which will add heads, and how to combine these two factors to kill each individual hydra in the shortest possible time.
The boss Mariska from Lollipop Chainsaw respawns into two when Juliet slices her in half. The copies then tear themselves in half several times to create an army of Mariskas. Juliet shoots the copies down, and when she gets to the last one, she lops her head off. Mariska seems to run out of energy and perishes.
Undead enemies in Wandering Hamster respawn every time they are killed, unless you use the glimmer item which kills them for good.
The Order of the Stick solved the problem of an actual hydra, by repeatedly decapitating it until it fainted - its heart couldn't maintain the proper blood pressure in all the extra necks.
Another episode had Triclops unleash an army of skeletal warriors that respawn into two every time they get smashed. They are defeated when the heroes smash the device Triclops was controlling them with, making them all crumble to dust.
Some weeds ensure their survival by sending out long horizontal roots that can sprout new stalks if the original stalk is destroyed. Pull up the first stalk, and you'll find a cluster of new weeds sprouting all around the hole a few days later.
Falling afoul of the Streisand Effect can land you in a PR nightmare because of this trope, especially on the internet. Did you force a takedown of a song posted to youtube, even though the way it was posted constitutes fair use? Expect much hate mail and at least fifteen reposts. Scoured the internet just to delete all references to you that you feel are overly critical, or connect you to something stupid/embarrassing you did? Congratulations, what you've censored has just been reposted to five blogs, and a sixth one commented on your dickery. Forced the takedown of an entire site? Well, now there's two mirrors registered under foreign domain names and five with slight variations of a common name.