The Lagrange inversion theorem is the essential tool needed to prove results like the following: Let $F(x)$ be the unique power series with rational coefficients such that for all $n\geq 0$, the coefficient of $x^n$ in $F(x)^{n+1}$ is 1. Then $F(x)=x/(1-e^{-x})$. For an application to algebraic geometry, see Lemma 1.7.1 of F. Hirzebruch, Topological Methods in Algebraic Geometry.

An alternating tree is a tree on the vertex set $\{1,\dots,n\}$ such that every vertex is either greater than all its neighbors or less than all its neighbors. Alternating trees arise in such contexts as the general hypergeometric systems of Gelfand and his collaborators, and in the combinatorics of the Linial hyperplane arrangement. Let $f(n-1)$ be the number of alternating trees on the vertex set $\{1,\dots,n\}$. Then $$ f(n) =\frac{1}{2^n}\sum_{k=0}^n {n\choose k}(k+1)^{n-1}. $$ So far as I know, the only known proof uses Lagrange inversion. (While this is a tree enumeration result, it is of a different nature than the standard applications of Lagrange inversion to tree enumeration.)