It's well-known that $e = \lim_{n\rightarrow \infty} (1+1/n)^n$ as defined by Bernoulli when considering infinitely-compounded interest. I believe this is the earliest definition of $e$.
But if we were in (say) the 17th century (before differentiation), how would we know that the limit exists and how could we calculate the value to arbitrarily many decimal places? Equivalently, how can we prove that $ e = \sum_{n=0} 1/n!$ without using $\frac{d}{dx}e^x = e^x$? (If we can prove $\lim_{h\rightarrow 0} \frac{e^h-1}{h} = 1$, that gives the derivative of $e^x$ and I'm fine with that approach too.)