  Let $G \leqslant {\rm Sym}(\Omega)$ be a finite almost simple primitive
permutation group, with socle $G_0$ and point stabilizer $H$. A subset of
$\Omega$ is a base for $G$ if its pointwise stabilizer is trivial; the base
size of $G$, denoted $b(G)$, is the minimal size of a base. We say that $G$ is
standard if $G_0 = A_n$ and $\Omega$ is an orbit of subsets or partitions of
$\{1, \ldots, n\}$, or if $G_0$ is a classical group and $\Omega$ is an orbit
of subspaces (or pairs of subspaces) of the natural module for $G_0$. The base
size of a standard group can be arbitrarily large, in general, whereas the
situation for non-standard groups is rather more restricted. Indeed, we have
$b(G) \leqslant 7$ for every non-standard group $G$, with equality if and only
if $G$ is the Mathieu group ${\rm M}_{24}$ in its natural action on $24$
points. In this paper, we extend this result by classifying the non-standard
groups with $b(G)=6$. The main tools include recent work on bases for actions
of simple algebraic groups, together with probabilistic methods and improved
fixed point ratio estimates for exceptional groups of Lie type.
