Here’s an unexpected application of exponential growth that I only learned about recently: the Gompertz Law of Human Mortality. It dictates that “your probability of dying in a given year doubles every eight years.”
Here’s the article that I read from NPR: http://www.npr.org/blogs/krulwich/2014/01/08/260463710/am-i-going-to-die-this-year-a-mathematical-puzzle?sc=tw&cc=share.