You will indeed need higher precision if you need a very large number of digits (more than what double precision gives you).
However, even with high precision, the algorithms you use to compute your moments matter: you might not have as many digits as you hope for. Variance, for example, is famously numerically unstable (sometimes giving a negative variance) when computed naively. A stable summation algorithm plugged into a variance computation (even a carefully written one) can sometimes give you six additional digits of precision in the variance that you might not get when doubling the precision (it depends on how big of a cancellation you had in your operations; too big, and doubling the precision will not be enough to recover, while a compensated summation would do fine).
I have seen this play out once in a clustering algorithm that would only work on some data when using a compensated summation to compute the variance.