Ouroboros

The symbol of a serpent eating its own tail, making a circle.

Dating back from Ancient Egypt, the Ouroboros is a symbol of eternity and the cyclical nature of time and the universe. The fact that the serpent is eating itself is also indicative of the idea of self-consumption in the way we live. Sometime it may even be two serpents each eating the others tail, taking on similarities with the Yin Yang symbol. Another, more modern common variation is for the snake or snakes to have an additional knot which forms the infinity symbol, ∞.

The Motif gains a lot of a secondary tier of association with alchemy and Time Travel. Note the connection to The Phoenix and also the serpent as the shedder of skin and rejuvenator which acts much like The Phoenix motif.

In some versions, the serpent encircles the world or the Tree of Life, and is tied to its fate.

Examples of Ouroboros include:

Anime & Manga

In Fullmetal Alchemist, all of the homunculi are marked for what they are by a tattoo of a serpent eating its tail somewhere on their bodies. Also interesting, is that "One is all, all is one" is sort of Arc Words for the series.
Crops up a lot in Noein, usually signaling that the amount of crazy quantum time travel hax is about to get even worse. The giant one that occasionally appears by a nearby island even serves as a portal to (effectively) The Legions of Hell near the end of the series.
In Mon Colle Knights, Oroboros is the master of the seventh realm, time, and as such it is an inorganic lifeform. It "appears" through the center of a gate that is the mathematical infinity symbol, and the Big Bad intends to use it to make everything one.
Tiger and Bunny features a crime syndicate called Ouroboros; members can be identified by ouroboros-symbol tattoos. For a while, Barnaby believed they were responsible for his parents' deaths, but it turned out to be a red herring.
- And while it seems like it is unrelated to the motif of the Ouroboros, in fact Ouroboros is fed by Maverick, who essentially leads the heroes. His actions cause Ouroboros to become a real shadowy organization as well, and they are essentially unstoppable, vowing to return at the end of the series.
The anime Souten Kourou—adaptation of Romance Of Three Kingdoms, but instead focusing on Cao Cao—starts with depiction of the Ton, a mega-massive grotesque snake-like/dragon-like/crocodile-like beast that eats the world. When it runs out of things to eat, it begins devouring itself starting from the tail, and eventually it itself ceases to exist along with all that it ate.
In Gunnm, Desty Nova has a Lotus Eater Machine he has named the Ouroboros program.

Comic Books

In "The Lord of the Mountains", an issue of Thorgal, a ring that allows time travel takes the form of an Ouroboros.
In Tintin in the Congo, Tintin makes a snake eat its own tail.
- It's apparently possible to create this in Real Life by feeding a snake's tail to itself. Snakes can't stop digesting once they've started, so this is a rather mean thing to do.
In the G.I. Joe Reloaded short Alternate Continuity, an ouroboros is the symbol Cobra carve in the place of one of their first attacks.
Mim, the queen of the dragons and creator of the living and dreaming worlds in the Bone mythos, is based on the Ouroboros. As long as her tail remains in her mouth, the world is safe. Then she gets possessed by the Lord of the Locusts...

Film

The Ferryman: Cheap horror flick involving an evil spirit which managed to escape death by possessing the bodies of the people on board a pleasure cruise. Every body it currently possesses bears a tattoo on their back depicting a snake biting its tail in figure of eight, an obvious symbol of eternity and immortality.

Literature

The Worm Ouroboros is a High Fantasy novel by Eric Rucker Eddison. The book is a story that is taken to be part of a wider conflict with an ill defined beginning or end which casts an interesting light on the heroics of the main characters. To reflect this we have the name and also the Book Ends of Here We Go Again.
The Wheel of Time series: Aes Sedai wear a ring shaped as the Great Serpent eating its tail. The most prominent symbol of the series is the Great Serpent looped in the shape of infinity, intertwined with the titular Wheel.
Discworld: Tethys, the water troll, landed on the Disc after falling off another world. One of the worlds he passed had a giant serpent eating its own tail around the disc, instead of the turtle/elephant arrangement of Discworld itself.
- This can be a reference or either the Norse mythology of Jörmungandr or, quite likely due to it also appearing in connection to the turtles-elephants-world model is Adisesha of Hindu tradition.
The Auryn in The Neverending Story is based on the two-serpent version.
- In the film, they make the specific addition of having the two snakes also intertwine in the centre to make the other common variation of having the infinity symbol in the centre.
This is mentioned in a crappy children's book called Redwork. It's about alchemy, and there's a ring in the shape of a serpent eating its own tail.
The above picture comes from the early alchemists' text The Chrysopoeaia of Cleopatra which dates all the way back to 2nd century Alexandria. "Chrysopoeaia" means rather literally "Making Gold" or basically, since we're dealing with alchemists here, transmutation into gold i.e. a Philosopher's Stone. You can also see from the light dark comparisons the similarity to the Taijitu.
In Robert E. Howard's Conan the Barbarian story "The Phoenix on the Sword", Thoth-amon's Ring of Power.

He triumphantly lifted a ring of curious make. It was of a metal like copper, and was made in the form of a scaled serpent, coiled in three loops, with its tail in its mouth. Its eyes were yellow gems which glittered balefully.

The ouroboros is an important recurring symbol in Wyrm.
In That Is All, John Hodgman explains that the ouroboros is not a symbol of eternity after all, but rather just a dumb snake that bit its own tail. Then it crushes Los Angeles. See also below under "Real Life."
The magical ring "Sning" in Incarnations of Immortality. It can come to life and it can answer yes or no questions by squeezing its wearer's finger.
In Septimus Heap, Septimus's Dragon Ring is an Ouroboros.
In Michael Flynn's Spiral Arm, the Ouroboros Circuit, a pre-human artifact, looks like this. Vaguely.

Live Action TV

In Millennium, members of the Millennium group are identified by the Ouroboros symbol.
The X-Files: Scully got an oroboros tattoo. Then she hooked up with a guy with a tattoo that talked to him.
On Red Dwarf, the episode Ouroboros reveals that Lister was discovered in a box with the word Ouroboros on it (originally misread as "Our Rob or Ross"). This turns out to be significant as Lister is his own father. The box originally contained Ouroboros-brand everlasting batteries.
Early on in The Invisible Man, Darien Fawkes gets a special Ouroboros tattoo that monitors the amount of counteragent in his blood via sections of the tattoo turning green or red. Once the entire tattoo turns red, Quicksilver Madness sets in.
Lost: In Mrs. Hawking's first appearance, she is wearing a brooch in the shape of an ouroboros. Appropriately enough, her role is to tell Desmond (who is in the past, but with memories of his future), to not deviate from the timeline he remembers. In Season Six, when Desmond meets her, she's wearing a different brooch that has two parallel lines on it.
On Doctor Who, the Doctor talks with Amy and Rory about a Time Lord colleague of his called the Corsair, who had an Ouroboros emblem tattooed on his (or sometimes her) body after every regeneration.

Music

The Smashing Pumpkins' fifth album Machina featured a lot of artwork from Vasily Kafanov and a lot of other alchemy inspired artwork. Yo can see quite a lot of it here and amongst the number, lies several of the Ouroboros.
The Bedlam in Goliath, the fourth full-length The Mars Volta album, contains a track named "Ouroborous" that loops back onto a pair of choruses and a series of guitar riffs that Word of God has declared to be a reference to this motif. Also fits in with the various weirdness that permeates the album, being as it is based around sayings and events that the band encountered while using an old ouija board they found in Jerusalem, and the string of unpleasant events that they thought were due to a curse from the board.
The Way of All Flesh the fourth album from French Metal band Gojira opens with the track "Oroborus" which describes the motif and its relationship with life and death. The album itself is about coming to terms with mortality and dying.
The anarcho-punk band Crass had an Ouroboros and a cross in their symbol.

Religion And Myth

As briefly mentioned above, Jörmungandr, the World Serpent, is the offspring of the giantess Angrboða and Loki who grew so large that he could encircle Midgard, our world, and grasp his own tail. When Jörmungandr lets go of his own tail, the world will end.
Adisesha is a primordial being in Hinduism who coils and forms the seat of Vishnu. Adisesha does not actually consume itself and it in fact has thousands of heads but still Adisesha forms a symbol for infiniteness and the endlessness of time.

Tabletop Games

In the Iron Kingdoms setting, the druids of Circle Orboros are named after Ouroboros (minus one o). They worship the Devourer Wurm, the embodiment of chaos.
The Legion symbol of the Thousand Sons Chaos Marines is an Ouroboros, in keeping with the motifs of the Marines themselves, and their cities.
Quite surprisingly for a Fantasy Kitchen Sink, Magic: The Gathering rarely has Ouroboros in its card arts. There are only two so far: This one is an alternate art of a card whose original art doesn't have Ourobors motif, and this one's art is that of an actual Ouroboros. Neither of them are named Ouroboros either, a situation which is a far cry from Phoenix, which are already pretty rare in Magic. At least the cards that have that motif in the art do have abilities that signify immortality.
In Vampire: The Masquerade, the Tzimisce clan symbol is a sinister rendering of an ouroboris. Several theories abound as to the nature of this fact, but the best explanation is that the clan is very Genre Savvy as to the cannibalistic nature of the Crapsack World they live in.
In Werewolf: The Apocalypse, Chronicles of the Black Labyrinth discusses the Ouroboros as a symbol of the Wyrm.

Video Games

City of Heroes has an organization called Ouroboros who seek to fix problems caused by people messing with time. In game terms, their facilities allow players to replay story arcs they have outleveled. Their symbol is a golden, angular ∞ version.
A recurring motif in Legacy of Kain which involves immortal consumers of blood and Time Travel.
Lost Kingdoms has one as a high level mon you can evolve one of your dragons into.
Ouroboros is the Sealed Evil in a Can in the game Brigandine
Hazama/Terumi (of BlazBlue fame) has a Drive called Ouroboros which grants him long range attack options. To further the thematic 'cycle' in Ouroboros, this guy is responsible for creating the time loop that plagues the first game. To further the thematic 'autoconsumptive' theme in Ouroboros, his weapon exacerbates emotional vulnerabilities in his enemies, sometimes allowing him to talk them into spiritually cannabalizing themselves.
Metal Gear Solid 2 has an Ouroboros appear in the map screen if you try to use it when GW is glitching. This hints at the purpose of the S3 program.
- Also shows up more subtly with the fact that the story's about a bunch of Snakes doing the same thing over and over and over and over and over and over and
The apocalyptic Virus threat in Resident Evil 5 is named "Uroboros", since it mutates its host into a giant mass of intertwining tentacles.
In Mega Man ZX Advent, the Ouroboros is a giant floating ring made up of all the Model Ws, which serves as the final level of the game.
The last boss fight of No More Heroes takes place in an arena surrounded by an energy dragon that eats its own tail. It constricts more tightly as the battle progresses, representing the self-destructive cycle of revenge.
The final boss of Xenogears is Ouroboros, and its defeat probably represents the end of the 10000-year cycle of death and rebirth Deus initiated in order to repair itself and fly off to destroy other planets.
In Shin Megami Tensei: Strange Journey, the boss of Sector Eridanus is Ouroboros. You have to fight her twice over, because at the first time, five demons are giving her energy to survive indefinitely. She's beaten at the second try, but who said that it'll be easy? Then it turns out that defeating Ouroboros doesn't actually enable the Red Sprite to escape the Schwartzwelt... She appears as a silver serpent, her body in a vertical infinity symbol, tail in her mouth.
In the Fallout: New Vegas add-on Old World Blues, all the Think Tanks have looping thought processes as a result of a limitation of their programming. To reflect this, they've all taken on names that represent some form of infinity, including Dr. Borous, shortened from Ouroboros. A smart character can call him on the misspelling, though he'll Hand Wave it. The other five are Doctors Klein (Klein bottle), 8 (infinity symbol on its side), Dala (short for "mandala," sanskrit for "circle"), 0 (a literal circle), and Mobius (Mobius strip).

Webcomics

Gunnerkrigg Court: In the third treatise, there's an ouroboros symbol in the sky. In a later page, a stylized depiction of Coyote appears to reference the ouroboros.
In this Unwinder's Tall Comics, there's an ouroboros in the panel border art. It's the first hint that this metafictional webcomic is doing something amazing.

Web Original

Mortasheen: Meet the Grobbidile. Doesn't really do much except eternally eat itself, though.
- And now, meet its cousin!
In the Strong Bad Email mini-golf, the "Worm Hole" dodges Homestar's putt, after which the Worm himself pops up, shouts "Get Your Own!" and takes a bite out of the golf ball. Then another Worm pops out of the golf ball, bites the first Worm, and the two end up forming an endless loop.

Homestar: "I'm gonna say that's a birdie. Yep...definitely a birdie."

Western Animation

Crom Cruach, one of the Old Gods from The Secret of Kells, combines this trope with Art Initiates Life, as this ancient deity who apparently wiped out The Fair Folk of Ireland before people stopped worshipping him takes the form of a Celtic Knot. Crom not only appears as an infinitely long serpent, he resembles a deep sea gulper eel.
- Brendan, the hero, has to steal the Eye of Crom using a Magic Crayon, whereupon Crom, blinded, consumes himself.

Real Life

Friedrich August Kekule von Stradonitz realized that the unusual properties of benzene could be explained if its carbon chain connected around in a hexagon. He claimed he got the idea after having a dream in which he saw a serpent eating its own tail.
Carl Jung, he of Jungian Archetypes, proto-troper, included this in his writings.
It is possible that the inspiration for the symbol is the view of the Milky Way in the night sky hence its appearances in astronomy focused cultures.
It's possible for real snakes to bite their own tails. Usually, the real-world ouroboros dies when this happens. This can happen if the snake is hungry and it smells the scent of some other creature on its tail. It will strike its own tail and attempt to devour itself - almost always with fatal results. Pythons, boas, rat snakes, king snakes and corn snakes have been photographed as ouroboroi.
Behold: Armadillo Lizard, a real-life Ouroboros. Even if the animal itself doesn't really symbolize anything.