At an eating contest you boast
That you can eat the most
How do you down your fiftieth piece of pie?
Eagerly (ugh!), eagerly (yech!), Eager L.Y.

-Tom Lehrer, L-Y

Exactly What It Says on the Tin, two characters, one of them possibly being a Big Eater, engage in this type of contest, where they try to outeat each other with as much food as possible. Often, the Big Eater will come out as the victor, while the other loses by becoming full too fast, sometimes ending up with a Balloon Belly.

There are also some cases where both characters will lose in the midst of the contest, by also gaining the aforementioned Balloon Belly. Also, after the Big Eater wins, sometimes he'll ask, "What's for dessert?" or show signs that he's still hungry.

Contrast Cooking Duel, although they may go hand-in-hand. Compare Drinking Contest.

Examples

Anime and Manga

JP and Zoe engaged in one of these in episode 17 of Digimon Frontier. Too bad that JP was the one who ended up with the Balloon Belly in the middle of it, while Zoe just kept on eating the food that came her way. Takuya and Koji even lampshade it with the following dialogue in the English dub:
Takuya: Zoe ate all that!?!? Where does she put it all?
Koji: Well, she is a growing girl.
The Picollet family of Ranma 1/2 practices Martial Arts Dining as one of the series' many Martial Arts and Crafts, with the goal not being to see who can eat the most, but who can eat the fastest, an area where the Picollets have a rather disturbing hereditary advantage. Additionally, Martial Arts Dining requires that one eats "elegantly", where "eating elegantly" means "chewing and swallowing so fast that it can't be seen by the unaided human eye". The only way to win against a Picollet is to shove your own food into the Picollet's mouth without him noticing.
Maid Latte puts on an eating contest in Kaichou wa Maid-sama!, with Supreme Chef Usui making the dishes, as a means of showcasing their dessert menu; Misaki participates in disguise as a favor to her co-worker Erika, who carelessly promised one of her regular customers that she'd go out on an all-day date with him if he won the contest. They both lose to bottomless pit Hinata.
Fuu enters an eating contest in one episode of Samurai Champloo. This earned her a surprising reputation as the Big Eater.
In Sket Dance, there is a chapter dealing with a ramen shop where the arrogant owner challenges the customers to completely eat his Biggest Damn Ramen (7 kg!) for a prize.
One episode of Magical Play involves Zucchini setting up an eating contest after overhearing Myumyu mention she's trying to keep her figure. Knowing how competitive she is, he figures she won't be able to resist, but doesn't count on her simply feeding all the food to her cats, who also function as her outfit. In the end, she wins, but when she stands up to accept her prize, her cats are too bloated to stay attached to her and fall, leaving her completely naked.
Lina and Gourry got into one in the fourth season of Slayers. The loser (The one to eat the least amount of food) had to pick up the check.
Naruto and Sasuke once compete by having an Eating Contest, finish at the same time and promptly throw up.

Comic Books

An Archie Comics story had Jughead compete against a member of a rival gang in a hamburger-eating contest at Pop's chocolate shop. Jughead won in the end, because no other Big Eater could quite match up to Jughead's style.
In the old Polish comic Koziolek Matolek, the titular goat encounters some indians, and after being challenged to a duel by their chief, the goat picks the competition to be eating grass. The chief eats until he explodes.

Literature

A fairly common competitive sport, albeit one forbidden by orthodoxy, in the Twenty Minutes into the Future Dystopia Thinner Than Thou. One of the protagonists is even a semi-professional participant.
The Doctor Who Expanded Universe short story "Too Rich for My Blood" by Rebecca Levine (in the collection Short Trips: Seven Deadly Sins, representing Gluttony) is set at a hot dog eating contest in Las Vegas. One of the contestants has been doped with an alien serum to increase his appetite. When they run out of hot dogs, he starts on the other competitors. And then it goes viral...
In Clarence Goes to Town, Clarence, a realistically portrayed dog, is in a quiz-and-stunt contest with a bunch of humans. One of the stunts is a pie-eating contest. The show runners decide that, since pie isn't good for dogs, Clarence's pie is substituted with a half pound of raw hamburger.

Film

In the live-action Underdog film, one of the scenes shown during Shoeshine/Underdog's bouts of crimefighting showed him in a hot dog-eating contest with a fat guy. Underdog beats him before the fat man could even get a bite.
Timon and Simba both competed in a bug-eating contest in The Lion King 1 1/2.
Inverted in Stand by Me, where Gordie tells a story he made up about a kid entering an eating contest just for the chance to drag his entire town into a mass vomit-chain as revenge for treating him as the Butt Monkey for years.
George and Beethoven enter a hamburger-eating contest at the county fair in the second Beethoven film. They won by only three burgers remaining.
Mr Woodcock has this.

Newspaper Comics

A Garfield strip had Garfield and Jon have a popcorn-eating contest together. However, Garfield won unfairly by eating both bowls of popcorn instead of one kernel at a time like Jon did. Another strip had Garfield and Jon have a hot pepper-eating contest to see who would endure the hotness of each different pepper they ate. Garfield lost in the end when he ate a Peruvian Death Pepper.

Live-Action TV

An episode of CSI: Crime Scene Investigation had a man who died from an over-expanded stomach in the midst of an eating contest, because his brain could not process 'full' signals from his stomach. This happened because the person who cared for him was a Jerkass who wanted to use him to win the contest.
- Also, because in the end, it was revealed that the man actually had a neurological disorder called Prader-Willi Syndrome, definite Squick.
The Drew Carey Show: Drew enters a pie-eating contest at a county fair and beats the reigning champion, a pig.
The Brady Bunch: When Bobby gets envious of the other Brady kids for all having trophies except him, he enters an ice cream eating contest on TV. He loses.
Malcolm in the Middle: When the family discovers that the neighborhood has a block party whenever they leave, Hal and Lois decide to show them up by winning the kielbasa eating contest.
In 30 Rock, Kenneth entered a pig-eating contest to win money to leave for New York to start his dreams. Turns out his pig was his old childhood pet Harold.
Kenneth: I ate all of him. Even the face in case of a tie. I ate my father-pig!
An episode of Weeds featured this. The food in question was butter.
On Yes Dear, Greg mentions in one episode that he and Jimmy once had a contest to see who could eat the most butter.
The Australian anthology series Swap Shop once had a story about an eating contest in which two Big Eaters competed to see who could eat a whole cow (converted into meat pies) the quickest. The local champion gained an early lead, but stopped before eating the last of his pies, giving the challenger a chance to catch up and then to win. When his friends asked him later why he'd stopped, he explained that he'd misunderstood and thought the pies were just an appetiser: he'd foregone the last pie because he wanted to leave room for the cow.
Has happened occasionally in the reality show Man vs Food.
Jim and Andy from According to Jim once participate in a hot dog speed-eating contest for the prize of being the spokesperson in a local commercial sponsored by the company of said hot dog product. Too bad for them they both lose... to a woman with smaller body. Jim's so unhappy about this that he challenges her to another speed-eating contest.
Bones had an episode where the victim was a competitive eater.

Tabletop Role Playing Games

In Warhammer, this is a form of Duel to the Death between ogres: The winner gets to eat the loser. The ogres will gladly take similar challenges from non-ogres, except trolls, whose Hollywood Acid digestive system is a Game Breaker even by ogre standards.
- In Warhammer 40,000, the Space Wolves Primarch, Leman Russ, went into an eating contest against the Emperor when they first met. He WON. Empy only got his allegiance by challenging him to a proper fight and winning.

Web Comics

In Men In Hats, Aram enters a taco eating contest to compensate for his lack of real emotions so he can have something to write in his diary.

Web Original

In the Whateley Universe story "Parents' Day", Tennyo's family comes to see her, and she ends up in an eating contest against her mother (both her parents are mutants too - so are her brothers).

Western Animation

An episode of Garfield and Friends had Jon and Garfield go to an authentic Chinese restaurant while visiting China. While there, the owner tells them the story of how thousands of years ago, a cat similar to Garfield outwitted the dragon out of all the food that he stole by bullying the villagers by challenging him to an eating contest. When the dragon discovers that he had been tricked out of all the food he acquired, he then chases the cat clear across China, making it sort of a subversion. When the story is finished, the owner then warns Jon that if an orange cat with black stripes (referring to Garfield) should ever eat too much whilst in China, the dragon will return and exact his revenge. However, much to the horror of both Jon and the restaurant owner, Garfield had already eaten all of the food that was in the restaurant while the owner was telling the story, claiming it to be a long fable, so he and Jon leave afterward.
The Simpsons: Homer gets in an eating contest with a trucker named Red, who dies right after beating Homer by finishing a 16-pound steak first. Just before the contest Red had eaten an entire lamb. Before the contest, Red and Tony Randall had been the only two people ever to complete the challenge.
Coop in Megas XLR competes in these, and has three championship crowns from the "Lord of the Large Pants" eating contest.
Arnold competes in one in the Hey Arnold! episode "Eating Contest".
A Ripped from the Headlines episode of King of the Hill featured a hot-dog eating contest where Dale shows up Bill by double-fisting the hot dogs, but losing in the end to a skinny Asian speed-eater.
In Jackie Chan Adventures, Jackie and his friends have to compete in an eating contest to find the tiger talisman.