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.
He's far from the only one. There's a reason the current time period is called "The Gourmet Age". All of civilization revolves around finding and eating delicious food. Justified since the Gourmet Age came right after and is immediately followed by a worldwide famine.
Goku from Dragon Ball Z. Justified in that he weakens a lot if he's hungry.
Again naturally, there was a one-hour Cross Over special that saw the casts of One Piece, Toriko, and DBZ competing in a giant foot race; since the prize was the tastiest meat in the world, of course, Goku, Luffy and Toriko were the most motivated runners.
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 Sangatsu 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.
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.
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."
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!'
Played for laughs in the song "But Now I'm Back" by Pink Martini. Lorenzo can't even focus on reconciling with his very peeved girlfriend, Maria — he interrupts his own plea for forgiveness by saying he needs to get a bite of grub with the boys.
Garfield is like this every time Jon puts him on a diet.
He's often like this even when he isn't on a diet.
In Nomine. Not surprisingly, this is true of Haagenti the Demon Prince of Gluttony.
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.
Taokaka from BlazBlue's always looking for food (this seems to be a trait of Kaka in general, but Tao's definitely 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 PokémonMunchlax. According to its Pokédex entry, it is so desperate to eat, that in its desperation it forgets that it has food stashed in its fur.
Appears to be a low-key Running Gag in the Golden Sun series. Looking in ovens in various towns will get reaction commentary apparently from the Heroic Mime protagonist du jour, Garet and Tyrell are mentioned to be Big Eaters, and Kraden's main concern about being set adrift at sea is lunch. To be fair, what we see of the oven contents is pretty dang enticing.
Manpuku from Ōkamiden clearly has some sort of eating disorder. He eats so much that there is never any food in the house and his mother skips meals often; and while he is with you, if you take the time to talk to the NPCs, most of his responces have him mention food from nowhere.
Jones in Criminal Case. He's asked by at least two other characters whether he ever thinks about anything besides eating.
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!"
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 Dexter's 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.
In the Woody Woodpecker short 'A Fine Feathered Frenzy' Woody reads a personal ad about a Gorgeous Gal who is rich, has lots of food and wants to marry a young man. Woody fantsizes about the food more than anything else. The woman had a sexy voice but turned out to be an old and overweight boy crazy bird. Upon meeting her, Woody didn't want her but she wanted to jump his bones and used his obsession to her advantage. When the Woodpecker goes to sit at a huge table of food he winds up on her lap. She pulls the turkey legs he is about to eat away from him and tries to grab him. Woody runs to the other end of the table, but with a push of a button marked 'Cozy' she transforms it into a small romantic table for two. The Woodpecker runs before taking a bite of food. 'Gorgeous Gal' also attaches a huge roasted turkey to a long string like a worm on a fishing hook to lure him into a make-out session. Gorgeous uses the turkey on a string again to lure him into a submarine where she traps him and somehow arranges for a preist to marry them.
The titular character of Chowder. Not surprisingly, he's an apprentice cook.