What if the simple interest isn't 100%? In other words, suppose your money wouldn't have doubled under simple interest, but increased by some other factor?
Suppose for instance the simple interest is 200%. Then we can split the time period up into two halves, with the simple interest being 100% for each half (for example, if you earn 200% simple interest per year, you're earning 100% interest for each six-month period).
At the end of the first half and the beginning of the second half, the balance under compound interest will be e times the original balance.
At the end of the second half, the balance under compound interest will be e times the balance at the beginning of the second half. In other words, it will be
e times (e times original amount)which is e^2 times the original amount (with "^" standing for "to the power of" on text-only browsers; if your browser can display graphics, switch to the graphical version for better notation). So, if the simple interest earned is 200% (2 times original amount), the final balance under compound interest is e^2 times the original amount.
This turns out to be true in general. If the simple interest earned is R times the original balance, then the final balance under compound interest is e^R times the original balance.
This is one of the reasons why exponentials of the form e^x occur more frequently in practice than do exponentials with other bases such as 10^x, and why one usually uses exponentials and logarithms base e rather than base 10 or any other base. (The other, more important, reason has to do with the special role that base e exponentials play in calculus).