Calcium carbonate, is the main ingredient in tums, and is the main component of limestone.

What a fascinating question! I hadn't previously considered this.

Baking soda is also an inorganic compound.

That our diet stems from mostly molecules described in organic chemistry can also be the direct result of the fact that there are vastly more molecules considered "organic": about 19 million are known and the number is growing!

While for anorganic compounds, there are only about 100,000.

The separation into organic and inorganic chemistry is really only done to make it easier to talk about broader subjects in science. We need and use obviously a lot of compounds that aren't carbon based large molecules.

Can we eat rocks, salts, metals, oxides… and I just don’t know that?

You'll die without Iron, Calcium, Potassium, Magnesium, Zinc etc. in your diet. Your body relies on a lot of metals and salts to function.

You can see an example of a food's breakdown here to see all the metals and salts involved in foods: Nutrition data for rice (make sure you scroll to the "Detailed Nutrition Data" section and expand the categories like "Minerals")

Fun fact, calcium supplements are Calcium Carbonate (CaCO3) - literally chalk. But for metals which are deliberately added to food, check out the Lucky Iron Fish developed to address anaemia in Cambodia.

It depends on what you mean by "eat." Does being able to survive them traveling through your digestive tract count as "eating" something? Does it have to have nutritional value? If the former, any inorganic substance with a low enough LD50 in a low enough dose would count. If it actually has to have nutritional value, youre limited to minerals like Calcium Carbonate (chalk and this isnt organic despite Carbon being in its composition), Potassium Chloride (no salt), Magnesium Oxide (milk of magnesia), Iron Sulfate etc. and any inorganic derivative that has relatively low toxicity.

Taco Bell.

A number of food additives don't contain any carbon atoms; if you follow the links from the Wikipedia article on food additives you can find a bunch, for example sodium bisulfate which is a color retention agent.

There are also some interesting cases like the antifoaming agent Polydimethylsiloxane (found in trace amounts in, among other things, Chicken McNuggets) which I believe is technically a hybrid organic/inorganic polymer.

Some people with too much money eat gold-plated food. So in a sense, gold is food, and it's not an organic molecule.

Edible clay is something that is actually sold as a product for people to eat. It's not metaphoric. It is exactly what it sounds like.

If you're asking if you could survive off only non-organic, than no. Your body won't generate energy from those things. Broadly fat, carbs, and proteins are the primary sources (there are others, but they can mostly be shoe-horned into one of those or they're insignificant sources).

You also need a better definition of "food." You can ingest a lot of things that are considered inorganic and not die, but that doesn't make them food. You'll still essentially die of malnutrition though. So your body will still starve, so I guess you could use that as part of the definition of food I suppose. But then salt doesn't count as a food anymore, neither does water (but that makes sense at least... water has never been considered a food really).

I think you're using a lot of words that have ambiguous meanings that folks don't usually think about and that's going to affect a lot of answers.

I’ve seen gold leaf put in food. It doesn’t hurt your body, apparently, but I guess you take nothing of value from it.

In recent years, activated charcoal.

(Because many people apparently don't know this: Don't eat activated charcoal if you take any medication, it can render your medication ineffective)

Edit: Wait, I'm dumb, charcoal is very much carbon-based.

Many minerals that are quite essential are used by other organisms so we get it that way, but that doesn't mean we have to. A good example is iron. Rust isn't a bioavailable form of iron compared to what you'd get from plants and animals, but you could eat rust and make some of the iron in it ferric chloride that you could then use for making hemoglobin. It's just not very efficient. What you can't do is get energy from inorganic molecules. We can only get energy from things on the citric acid cycle(or are made up of components of it), alcohols, sugars, fats, and proteins.

Hydrochloric acid, sodium hydroxide: common food additives to balance its pH. Technically included by the OP, as they're made from water and salt.

Silica: another common food additive (anticaking agent). Arguably "not food" because your body will simply poop it, undigested.

Metallic iron: if you're cooking in a metallic pot, some of the metal leaks into the food. Actually good in this case as iron is an essential mineral.

Metallic lead [historical]: the Romans used lead pots quite a bit, specially to boil either vinegar or barely fermented wine (mustum). The result is extremely sweet due to lead acetate, the first artificial sweetener that I'm aware of ...and extremely toxic. (OK, it's a partially organic salt due to the acetate anion, but still.)

Calcium carbonate: already mentioned by another user. It's essential to nixtamalise maize, but it's also used for stuff like candied pumpkin cubes.

Bicarbonate soda, used heavily in baking. Vinegar (acetic acid as opposed to natural fermentation)