  In this paper we give a bound to the number of conjugacy classes of maximal
subgroups of any almost simple group whose socle is a classical group of Lie
type. The bound is $2n^{5.2}+n\log_2\log_2 q$, where $n$ is the dimension of
the classical socle and $q$ is the size of the defining field. To obtain the
bound, we first bound the number of projective cross-characteristic
representations of simple groups of Lie type as a function of the
representation degree. These bounds are computed for different families of
groups separately. In the computation, we use information on class numbers,
minimal character degrees and gaps between character degrees.
