Food is important. Without the many things it provides (not just nutrition, but culture, comfort and, ...well, just gives life more flavour) people would just keel over and die (...literally). Of course, this makes being hungry a big deal.
With this in mind it's hardly a surprise that some people just can't get it off their mind. Not just in the sense that they don't want to starve, but that they can't stop thinking about eating. They might just constantly crave their Trademark Favorite Food, or just food in general. Either way, they will pay more attention to food than anything else. Expect another character to ask them "How can you think about food at a time like this!?" at some point or another.
Most of the time, especially in Western Animation or Anime and Manga, it's Played for Laughs, but it can also be Played for Drama.
These characters tend to be the Big Eater and may suffer from Meat O Vision and are prone to a Delicious Distraction. Compare Inappropriate Hunger, which is often overlapped with this when Played for Laughs. Contrast Weight Woe, where someone's trying to ignore food.
Examples:
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Anime and Manga
In Trinity Blood, Played for Laughs with Father Nightroad at the start of the first episode of the Anime, who hasn't eaten in 22 hours due to not being able to pay for a sandwich. Later, it's Played for Drama with Ion, who is a Friendly Neighborhood Vampire afraid of losing his self-control. However, when they're able to eat normally, neither is obsessed with food.
In One Piece, Luffy gets hungry real easily to where he will try to eat the nearest thing as a Running Gag. He particularly likes meat (of any sort) but will eat anything that tastes good. After a few hours without eating, he will start to complain to anyone nearby. After several hours, it substantially affects his ability to fight—normally Played for Laughs but becomes a major plot point when he fails to stop the CP9 agents from assassinating Iceburg, burning down his mansion, and kidnapping Franky.
Toriko doesn't think about anything but eating (justified since he has a Hyperactive Metabolism which means he'll starve in hours rather than days and gathering ingredients and cooking's kind of a Serious Business in his world). Naturally the anime adaptation has crossed over with One Piece.
Goku from Dragon Ball Z. Justified in that he weakens a lot if he's hungry.
Sakura Kyouko from Puella Magi Madoka Magica is rarely seen without chewing some kind of food. Normally, something like that doesn't qualify her for this trope, but it's revealed that it's almost entirely psychological, due to her upbringing as a poor child and the fact that magical girls actually don't need to eat. That's an obsession if there is ever one.
In 3-gatsu no Lion, all Akari's cats can think about most of the time is food, who's going to give it to them, and when it's going to come.
Comic Books
In Archie Comics and the animated series thereof, Jughead Jones would often become distracted if the subject of food came up in the middle of him talking about something else.
Literature
In Taft 2012, the protagonist lampshades this, describing his interest in politics as a "hunger for values".
Augustus Gloop, finder of the first golden ticket, who is constantly eating. Possibly the inspiration for The Simpsons' Uter.
Charlie Bucket and his family are also preoccupied with food but for a different reason: they're so poor that they're starving.
In The Hobbit Bilbo is constantly thinking about where he'd rather be and what he'd rather be eating (usually cake). This is also pretty much the Shire's hat.
Each of the Children in Galaxy of Fear: The Hunger are this to some extent, because there's just not much they can find that's edible on Dagobah. They're all pretty malnourished.
"Eat." Galt whispered the word as if it were a secret, magic spell. His eyes bored into Zak but seemed to look through him. "We eat what we can. We eat when we can. Always hungry. Always. Mostly we eat fungus."
Live Action Television
In an episode of The X-Filestold from the Monster of the Week's POV, said monster is consumed with eating people. He tries to control it by going to Overeaters Anonymous, but ends up killing & eating his landlady (who he bonded with when he discovered she was also in OA).
On the first series of Skins, Cassie thinks about food a lot. She has an eating disorder, she organizes tinned food in other people's homes, has set strategies for pushing food around her plate to make it look like she's been eating, and believes someone is sending her secret messages telling her to 'EAT!'
In Nomine. Not surprisingly, this is true of Haagenti the Demon Prince of Gluttony.
Video Games
Soul Calibur V has Xiba, a character who is shown eating in his promo art and is so concerned about getting his next meal that his opening battle lines include, "If food is on the line, I'd better win this!" and "So if I beat you, you'll give me some food?" He will even yell "Food!" when knocked out.
Bud from Mega Man Star Force is obsessed with food to the point that him entering an eating contest is a plot point. Obviousley, Played for Laughs.
Taokaka from BlazBlue's always looking for food (this seems to be a trait of Kaka in general, but Tao's definatly the most obsessive). She even nicknames Ragna the Bloodedge "Good Guy" (even though she was supposed to capture him for a bounty) because he fed her. Similarly to Xiba (from Soul Calibur, mentioned above) he battle quotes often reference food (up to and including the possibility of eating her opponent).
The Gaping Dragon of Dark Souls was originally a normal stone dragon that became so obsessed with eating its ribcage turned into a giant maw.
Ilyana from the Fire Emblem Tellius series. Strangely enough, this is barely noticeable in the main plot of the first game outside of the epilogue despite all of her support conversations being about wanting to eat—at least by level A. This was made more obvious in the sequel, with practically every line she has being about food, and her ending plaque says "somewhere, right now, she is hungry."
The PokemonMunchlax. According to its Pokédex entry, it is so desprate to eat, that in its desperation it forgets that it has food stashed in its fur.
Visual Novels
Saber from Fate/stay night. Food is just very important to her. Not just any food, mind you. She likes good food. Remembering the highly inadequate skills of the chefs of her era has her briefly lets off a killing aura.
Saber: "This is your last warning, Shirou... Please make me lunch right now!"
Web Original
The Nostalgia Critic is a skinny bugger, but can be bribed with sweets and sees nothing wrong with a side order of liposuction fat for a hot-pocket.
In the Dexters Laboratory episode "The Bus Boy" when the fat German kid is telling his story about the creature at the back of the bus, he keeps distracting himself whenever he mentions food.
In the Hey Arnold! episode "Best Friends" when Arnold and Gerald are trying to find out why Rhonda and Nadine split up and they question Harold, he can only remember what they were eating and anything they said that was related to food.
Homer Simpson. Whenever he hears (or thinks he hears) any mention of food, he responds with the Catch Phrase "Mmm, [food/thing that vaguely sounds like food]", then spaces out and drools.
Uter the exchange student from Germany. Always seen eating chocolate bars.
South Park: Eric Cartman is obsessed with cheesy poofs and fried chicken.
In Scooby-Doo, Played for Laughs with Shaggy and Scooby. Both of them are slender, cowardly Big Eaters who are only not hungry when they've stuffed themselves with more food than would normally feed a dozen people, and are only not thinking about food when they're too scared.
Krumm of Aaahh!!! Real Monsters is an Extreme Omnivore to boot. When Ickis asked whether he thought about anything besides food, he thought for a moment and offered, "Beverages?"
Mr Bogus often showed shades of this most of the time, because of his Big Eater status.
Real Life
People suffering from Prader-Willi syndrome. These people think about food 24/7 because full signals never reach the brain, and as a result, they feel hungry perpetually.
King Louis XVI seemed more interested in food than his wife.
People who have been through famines are often obsessed with food, even long after the crisis is over.
