Your gut is pretty adept at moving things along but hypothetically, if it did get stuck, could we digest its different constituents?
The digestive system isn’t like an in-tray, where everything remains until fully processed: it’s more like a conveyor belt. Whatever you eat moves through your intestines at roughly the same speed, and anything that doesn’t get broken down and absorbed into your bloodstream passes out the other end. This usually takes one to three days.
Chewing gum can’t stick to the wet intestinal wall, so a single piece of swallowed gum normally gets swept along with everything else. There are a few cases in the medical literature of small children swallowing many pieces of gum that formed a lump too large to pass. But this quickly caused severe constipation and pain, and needed surgery to remove.
But supposing some gum did somehow get trapped, would it take seven years to digest? Chewing gum is 70 to 85 per cent sweeteners, flavourings and starch, all of which are digestible. But the remaining 15 to 30 per cent is a blend of synthetic polymers, often including butyl rubber.
This rubber is also used to make the seals on chemistry lab flasks, and is rated as suitable for storing hydrochloric acid that is 30 times more concentrated than the acid in your stomach. So if you could somehow withstand the severe constipation, the gum would likely last at least seven years.