They will eat anything edible and have survived in deserts eating cacti and in the Antarctic, stealing penguin eggs. In fact they will even eat paper, plastic, soap, metal, brick, wood...is it any wonder they're Explosive Breeders?
It must be noted, however, that rat teeth grow at a constant rate and must be constantly used to grind them down. Not everything that rats chew on is actually eaten — rats in captivity will reduce wood to sawdust without swallowing it.
Raccoons who live in cities will eat almost any food they can get, since they often have to scavenge for it.
Sharks tend to bite first and ask questions later, whether it's their normal prey or not. Played with in Jaws, where a dead shark's stomach has various junk in it, including a license plate.
Tiger sharks will eat anything. License plates have been found in the stomachs of actual sharks, along with tires, pieces of medieval armor, etc, etc.
Used tragically in Open Water, when a couple of fishermen find a discarded camera left by the unfortunate water-stranded couple in a tiger shark's stomach.
Criminally-neglected dogs will swallow rocks, sticks, and dirt in an attempt to ease hunger pains.
Many dogs have been known to eat their own vomit. Or poop. Or basically anything they can put in their mouths. From an evolutionary standpoint, eating their own waste was a way to keep predators from following them.
And then there's the thought process that many dogs have: "Mmm, this smells like food! (OM NOM NOM) Ew, that was gross! (BARF) Hey, what's this? (sniffs its vomit) Mmm, this smells like food!" Repeat ad nauseum.
Many species of catfish will eat anything that they can fit in their (rather large, for body size) mouth. Aquarium keepers have to keep this in mind if they intend to keep catfish, as any smaller fish get eaten whole.
Hyenas in general, and it's not just bones and fur they take, there have apparently been reports of them dining on aluminium cookware (though they later regurgitate the pellets).
Tasmanian devils. Objects found in this creature's stool include bones, eggshells, feathers, and even items scavenged from households, including part of a boot. Hopefully there weren't any toes in it....
Microbes in general. If you name a material, something actually does, or has the ability to evolve to be able to, eat it. For example, Halomonas titanicae eats iron, and its colonies can be seen growing on metal shipwrecks in the deep ocean, such as the Battleship Bismarck and the RMS Titanic.
Pigs aren't averse to chewing on clothes either, and even well-fed pigs will chew on stuff like straw, bark, or paper out of boredom.
Porcupines will attempt to eat pretty much anything with salt in it, up to and including leather work gloves or wooden tool handles that humans have sweated on. They also consume bones for calcium or fallen-off deer antlers for protein.
A Frenchman known as Monsieur Mangetout (French for "Mr. Eat-Everything") has eaten, among other things, 18 bicycles, several TV sets, a coffin and even an airplane. There was a subversion, however: hardboiled eggs and bananas, considered normal food, made him sick.
Charles Domery (or possibly Domerz), a French soldier was taken prisoner by the English in 1798; while in jail, in addition to a daily intake of ten men's rations, he also consumed a number of rats and candles. Before that, in the battle he was captured in, a fellow sailor's leg was blown off by a cannon; somebody had to yank it away and throw it in the ocean before he'd get back to fighting the battle. His favorite food was raw bull's liver. And when no other food was available, he would resort to eating kilograms worth of grass.
Tarrare (real name lost to history), was another Frenchman and Domery's contemporary, who ate, among other things, live cats, huge banquets meant for 15 people, corpses from the morgue, and one time a 14-month-old toddler. Incredibly, despite all the stuff this guy ate, he was supposedly of normal build and stature.
Steve, proprietor of "The Sneeze" ("Half zine. Half blog. Half not good with fractions.") has an entire category of posts devoted to his extreme gastronomy: "Steve, Don't Eat It!" You've got to admire a guy who'll eat such things as pickled pork rinds, Beggin' Strips? (a dog snack marketed in the US), natto, and canned silkworm pupae, and drink such things as his wife's breast milk and homemade "prison wine" in the interests of humor.
Most of those seem pretty tame compared to some of the things they eat in the TV shows above. Heck, the whole purpose of human breast milk is to be consumed by humans; cow milk is actually squickier, if you think about it a bit.
Benchilada, creator of "So You Don't Have To" will also consume all manner of bizarre food and drinks, so you don't have to.
Vitamin deficiencies associated with pregnancy can give rise to a temporary craving for materials rich in calcium and other minerals, such as chalk, clay, or even newsprint.
Some kids will pull and eat their hair. Ditto with younger infants, who are major Extreme Omnivores (or at least like to put everything in their mouths to determine what it is) — which is why they should never be left with things that could potentially fit into their mouths.
Coprophagia: Desire to eat feces. Child serial killer Albert Fish is a notable afflictee. Truthfully, feces eating should be left to bugs such as the dung beetles...
And then there's pica, a serious eating disorder in which sufferers habitually eat non-nutritive things like clay, paper, chalk, glass, wood, metal, et al. Eating metal can lead to perforation of the GI tract and usually death.
One man lost at sea developed cravings for normally-discarded parts of fish, such as eyeballs (for their water content) and internal organs (for vitamins).
Human defense cells called "macrophages" (a name which literally meansBig Eater) have the job of consuming any and all microscopic things that don't belong in a healthy body. They'll perform phagocytosis on anything, from pathogens, to dead or dying cells, to minute fragments of splinters, to toxic molecules of snake venom or atoms of arsenic.
Human beings as a species have a remarkable dedication to eating things nature either never intended them to eat or tried its darnedest to make inedible. In addition to being the only mammal that consumes the milk of other mammal species, we eat things like chili peppers (evolved to be repulsive to mammals with their spicy flavors), potatoes (originally poisonous, yet mankind still domesticated them), blowfish (incredibly toxic in all but a few select slivers of flesh), and several other varieties of poisonous or formerly-poisonous (before domestication) plant.
The fact that we're the only animals which cook our food also broadens our diet immensely, as we're equipped to extract edible bits from otherwise-unpalatable sources and process it into something digestible.
Humans will not only hop between food chains at will, they'll even hunt and eat the animals at the top of those food chains, such as crocodiles, sharks, and bears. Humans' intelligence and tool use allows them to kill and eat creatures that any other predator would see as too much work, and we're the only species with enough imagination and free time to try to eat some new animal just because we've never eaten it before, and wonder if it would taste good or not.
Humans are remarkably tolerant to natural plant defenses and poisons as well (which is why you should never feed anything meant for human consumption to a dog or cat unless you know for sure that it is safe!). Most spices come from defenses plants evolved to disgust, sicken, or even kill herbivores; the pungent flavors were a warning to the animal that it was eating something dangerous and should spit it out. By animal standards humans can consume massive amounts of caffeine and theobromine (a chemical found in chocolate) with no ill effect, and humans are one of the only animals that can safely consume significant amounts of alcohol. Other human foods that can hurt or kill pets include avocado, macadamia, bread (the yeast is bad for pets' digestive systems), artificial sweeteners (a healthier alternative to sugar for humans, a deadly liver toxin for pets), and garlic.
This Cracked article. People featured in it include a man who ate weeds from his garden, a soldier who ate rocks, a roman emperor that ate bird brains and gold, and to top it off, a scientist who ate a human heart.
YouTube user Shoenice has built a following for eating things like glue, deodorant and pencil erasers, among others.
In one of the few mechanical examples of this trope, the jet turbine. Essentially, if it's a flammable liquid you can atomize, it'll run quite happily on it (though not as well as it's intended fuel.)
The steam engine. Requires reconfiguring, but there's ways to make them run on coal and coal gas, oil, kerosene, propane, gasoline and diesel fuel, natural gas, sunlight, radioactive material, plant matter and charcoal, garbage and compost... Really, anything that gives off heat or has carbon in it.