whenever a process evolves at a speed proportionnel to its own advancement, it satisfies the differential equation $y'=a y$ which leads to exponential solutions.
this is a very basic model for all kind of population growth (as you'd say that the bigger the population is, the more baby they make per time unit).
it is universal in the sense that if $y$ satisfies a more complicated evolution equation $y'=f(y)$ near an equilibrium point (for example, if you have a very small initial population, or a starting process) then linearizing this equation you obtain something like $y' \approx f'(y_0) y$ where $y_0$ is the equilibrium point.
Addendum : radioactive decay follows a decreasing exponential law because of the time-independance nature of the probability for a particle to split.
Basic equations in kinetic chemistry are exponential laws (though they get more complicated for complex reactions), I suspect mainly because of the universal property I mentionned : it is "the most simple evolution to an equilibrium state" in a sense.