You might've heard the old wives' tale about how swallowing chewing gum is bad because it'll stay at the bottom of your stomach forever. While swallowing chewing gum isn't particularly good for you, the science shows that it probably won't just sit in your stomach. So then, what does happen?
Your body is designed to move things in and out of your digestive tract as efficiently as possible while squeezing out every nutrient that it can. But unlike most of the foods you might consume throughout the day, gum doesn't break down, so you won't get any nutrition from it.
Nevertheless, the muscles throughout your digestive system will eventually move any gum you swallow to the end of your digestive tract. Furthermore, it'll emerge undigested, much like certain parts of corn or sunflower seeds after a bowel movement.
While swallowing your gum once or twice probably won't be the end of the world, we wouldn't recommend doing so in significant quantities. Consuming too much can eventually cause digestive system blockages, which could require emergency medical attention.