Let $(X,\mathcal{F})$ be a locally ringed space. In a bonus exercise last year, we were asked to compute the completion in general of such a stalk on a smooth manifold of dimension $n$ (it is isomorphic to the ring of formal power series over $\mathbb{R}$ in $n$ unknowns). It's clear that this is a bad case to work with, since smooth manifolds admit bump functions (which allows us to prove that there exists a nonzero element in the intersection of all (finite) powers of the maximal ideal), and therefore the completion contains very little data.

However, what kind of "stuff" does this technique allow one to do in the analytic/holomorphic cases? Similarly, in the algebraic case, we should look at the henselization or strict henselization, but I still am not really sure why one would want to do so in the first place. That is, what geometric idea corresponds to the idea of "completion" (including henselization and strict henselization depending on the context) in the same way that localization at a prime corresponds to taking stalks geometrically?