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.
