Chewing gum is basically plastic doped with flavours and colourings. It is non-digestible and water insoluble, which means you can carry on gnashing away at the stuff and it never seems to break down. This indestructible property comes from the gum base, the exact ingredients of which are usually a trade secret. However they generally contain:
Fillers: Calcium carbonate or magnesium silicate (talc) provide texture and bulk. If the gum has acidic flavours, then talc is used. This is because calcium carbonate would react and produce carbon dioxide gas.
Elastomers: These are long polymer molecules with elastic properties. Until WWII, chewing gums used a natural latex derived from sapodilla trees, but since then synthetic elastomers, such as polyvinyl acetate, are preferred.
Emulsifiers: These help to keep other ingredients, including flavours and colourings, nicely mixed and also impart some anti-stick properties.
Softeners: Compounds such as vegetable oil and lecithin are added to the gum base to keep everything soft and chewy. Masticate for too long and these can get washed away, leaving you with an overly stiff piece of gum.