Enumerative geometry is the study of geometric counting problems. A major insight of the 1990s is a deep relationship to integrable systems. This manifests more precisely in the surprising observation that for many enumerative invariants, their generating functions provide solutions for integrable hierarchies. For curve counting with rational target, this relationship has been conceptualised in recent years using tools from various different fields spanning from combinatorics to mathematical physics. In this talk, we give an overview of some of the results and technique used. In particular, we illustrate how recursive combinatorics in tropical geometry interact with Chekhov–Eynard–Orantin topological recursion in the case of leaky Hurwitz numbers arising from logarithmic Gromov–Witten theory. This talk is partially based on joint work in progress with Reinier Kramer.