Non-D-finite excursions in the quarter plane

We prove that the sequence $(e^{\mathfrak{S}}_n)_{n\geq 0}$ of excursions in the quarter plane corresponding to a nonsingular step set $\mathfrak{S}\subseteq\{0,\pm 1\}^2$ with infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. Accordingly, in those cases, the trivariate generating function of the numbers of walks with given length and prescribed ending point is not D-finite. Moreover, we display the asymptotics of $e^{\mathfrak{S}}_n$.