Recent progress on character bounds for groups of Lie type makes it feasible in many cases to find the asymptotic growth, for fixed q and n tending to infinity, of the number of n-dimensional representations of a Fuchsian group G over the field with q elements. Applications include determining the dimensions of complex representation varieties of Fuchsian groups in high degree and proving the existence of strongly Zariski-dense Fuchsian subgroups, in the sense of Breuillard-Green-Guralnick-Tao, of classical groups of high rank.