We show a first-order asymptotics result for the number of galled networks with $n$ leaves. This is the first class of phylogenetic networks of {\it large} size for which an asymptotic counting result of such strength can be obtained. In addition, we also find the limiting distribution of the number of reticulation nodes of a galled networks with $n$ leaves chosen uniformly at random. These results are obtained by performing an asymptotic analysis of a recent approach of Gunawan, Rathin, and Zhang (2020) which was devised for the purpose of (exactly) counting galled networks. Moreover, an old result of Bender and Richmond (1984) plays a crucial role in our proofs, too.