From this paper.
Examination of the regrouped series for $e$, given below, shows what is the well-known method of rewriting polynomials in a slightly different guise:
$$1 + 1\left(1 + \frac{1}{2}\left(1 + \frac13\left(...\left(1 + \frac1n\right)...\right)\right)\right)$$
How do I rewrite this as a function of $n$ in big-pi notation?
I don't think there is a way to write this using $\prod$ notation, simply because it can't be broken up into a single, long product. All those multiplications that happen happen inside other factors, not next to other factors.
However, if you expand the parentheses, you can make it into a combined form with $$ \sum_{n = 0}^\infty\left(\prod_{k = 1}^n\frac1k\right) $$ which is basically just another way of writing the very standard $\sum_{n = 0}^\infty \frac1{n!}$.