We prove lower bounds on the growth of certain filtered Hopf algebras by means of a Poincaré–Birkhoff–Witt type theorem for ordered products of primitive elements. When applied to the loop space homology algebra endowed with a natural length-filtration, these bounds lead to lower bounds for the number of geodesic paths between two points. Specifically, given a closed manifold $$M$$ M whose universal covering space is not homotopy equivalent to a finite complex and whose fundamental group has polynomial growth, for any Riemannian metric on $$M$$ M , any pair of non-conjugate points $$p,q \in M$$ p , q ∈ M , and every component $${\mathcal C}$$ C of the space of paths from $$p$$ p to $$q$$ q , the number of geodesics in $${\mathcal C}$$ C of length at most $$T$$ T grows at least like $$e^{\sqrt{T}}$$ e T . Using Floer homology, we extend this lower bound to Reeb chords on the spherisation of $$M$$ M , and give a lower bound for the volume growth of the Reeb flow.