Hydra Problem

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.

—Lore Sjöberg, The Book of Ratings

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.

Examples of Hydra Problem include:

Trope Namer comes from the Lernaean Hydra from Classical Mythology. Whenever a head was lopped off the beast, 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 Cerebus (Hades' dog, not the aardvark), who it turned out had the same power. Samurai Cat just kept chopping off heads until Cerebus had so many, 1) they were all too small to bite effectively and 2) they weighed so much Cerebus couldn't walk around anymore.
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.
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.
Dungeons & Dragons 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.
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).
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.
Madrox the Multiple Man of the X-Men has this as his main trick, spawning copies each time he is hit. The answer to beating him is employing a method that doesn't involve direct force, such as poison.
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.
- Then an enterprising goblin starts using it as an infinite supply of meat for a barbecue restaurant.
A bit of a silly example, but a Oral Tradition example: That Old Wives' Tale about how picking a gray hair will cause two more to grow in its place.
- More Oral Tradition: Shaving makes your beard grow even fiercer.
In the Percy Jackson and The Olympians movie, they fight a literal Hydra (who has this problem). They beat it by using Medusa's head to turn it to stone.
- 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.
Undead enemies in Wandering Hamster respawn every time they are killed, unless you use the glimmer item which kills them for good.
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 rockslide.
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."
He-Man once faced a Man-Eating Plant that was similar to the Hydra. He finally kills it by uprooting it.
- 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.
Religion example: 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.
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.
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 Powerpuff Girls episode "Beat Your Greens" had Plant Aliens that regenerate. They are defeated when everyone eats them.
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.
Hydra Slayer is a quirky little Roguelike 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.
Real Life example: 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.
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.