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There is something called tonsillar stones or tonsilloliths. When we are born, the tonsils are small and have indentations in them. As we grow, so do the tonsils. The indentations for most of us get naturally smaller as the tonsils get larger. In the unlucky ones, they do not. Then partially decayed food, saliva, etc. collects in these crypts. They may occasionally notice that they cough or gag up small rounded semifirm bits of material that smell quite unpleasant. These are the "stones." Some people can aggressively gargle and remove them or flush them out somehow. Tonsillectomy is an option, but unless deemed medically necessary, insurance would not likely cover it.
You might be surprised to find this website that is dedicated to the very problem!