To a spectral curve $S$ (e.g. a plane curve with some extra structure), topological recursion associates a sequence of invariants: some numbers $F_g(S)$ and some $n$-forms $W_{g,n}(S)$. First we show that $F_g(S)$ grow at most factorially at large $g$, $F_g = O((\beta g)! r^{-g})$with $r superior at 0$ and $\beta\leq 5$. This implies that there is a Borel transform of $\sum_g \hbar^{2g-2} F_g$ that is analytic in a disk of radius $r$.The question is whether this is a resurgent series or not? We give arguments for this, and conjecture what are the singularities of the Borel transform, and we show how it works on a number of examples.