  Let $C$ be a smooth curve. In this paper we investigate the geometric
properties of the double nested Hilbert scheme of points on $C$, a moduli space
introduced by the third author in the context of BPS invariants of local curves
and sheaf counting on Calabi-Yau 3-folds. We prove this moduli space is
connected, reduced and of pure dimension; we list its components via an
explicit combinatorial characterisation and we show they can be resolved, when
singular, by products of symmetric products of $C$. We achieve this via a
purely algebraic analysis of the factorisation properties of the monoid of
reverse plane partitions. We discuss the (virtual) fundamental class of the
moduli space, we describe the local equations cutting it inside a smooth
ambient space, and finally we provide a closed formula for its motivic class in
the Grothendieck ring of varieties.
