This is unlikely a research level question... one that would be answered in a blink of an eye, rather...it is an (early) exercise from the book "Analytic Pro-p groups". But since no reply was received when posted on the sibling site, I guess I might try my chances here, risking an immediate closure of the thread (nope, not homework...)
Give an example of a finitely generated pro-p group $G$ and a dense subgroup $H$ of $G$, with $H$ finitely generated as an abstract group , such that $\hat{H} \ncong G$.
$\color{grey}{\rm edit}$: link to the other question: https://math.stackexchange.com/questions/495014/a-dense-subgroup-with-completion-not-isomorphic-to-the-big-pro-p-group