By iterating the type of pullback constructions in which $P^rVD$s arise by Nagata composition, we are led to study a class of inverse limits $A=\underleftarrow{\lim}A_n$ of integral domains indexed by $\boldsymbol N$. After identifying the prime spectrum, the localizations, and the integral closure of $A$, we then characterize when, i.a., such (typically infinite-dimensional) $A$ is a Prüfer domain, Bezout domain, divided domain, or $P^rVD$.
