How does the story of two ancient kings illustrate exponential growth?

The story of the two ancient kings is a good illustration of exponential growth. This can be illustrated through their chess game where the king always doubled the amount of gain that he placed on each square of the chess board. Before doing so, the king realized that he did not have enough gain to double it on each square.

This clearly shows that an exponential growth might start out very slowly, but speeds up and increases very quickly.