  Given a local noetherian ring $R$ whose formal completion is integral, we
introduce and study the $p$-radical closure $R^\text{prc}$. Roughly speaking,
this is the largest purely inseparable $R$-subalgebra inside the formal
completion $\hat{R}$. It turns out that the finitely generated intermediate
rings $R\subset A\subset R^\text{prc}$ have rather peculiar properties. They
can be used in a systematic way to provide examples of integral local rings
whose normalization is non-finite, that do not admit a resolution of
singularities, and whose formal completion is non-reduced.
