  The number of excursions (finite paths starting and ending at the origin)
having a given number of steps and obeying various geometric constraints is a
classical topic of combinatorics and probability theory. We prove that the
sequence $(e^{\mathfrak{S}}_n)_{n\geq 0}$ of numbers 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 $e^{\mathfrak{S}}_n$.
