Why does gum get harder to chew after a while?
I would be surprised if there was reticulation of the gum's polymers upon contact with your saliva. According to this manufacturer, plasticizers are used, and the plasticizers appear to be fats. This would mean that, upon chewing, the plasticizers would be removed from the gum, rendering it unpleasantly harder.

Plasticizers are used to render a polymer more pliable, more elastic, than it naturally is. This could get its own question, imo.