  Let $G$ be a finite primitive permutation group on a set $\Omega$ with
nontrivial point stabilizer $G_{\alpha}$. We say that $G$ is extremely
primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega
\setminus \{\alpha\}$. These groups arise naturally in several different
contexts and their study can be traced back to work of Manning in the 1920s. In
this paper, we determine the almost simple extremely primitive groups with
socle an exceptional group of Lie type. By combining this result with earlier
work of Burness, Praeger and Seress, this completes the classification of the
almost simple extremely primitive groups. Moreover, in view of results by Mann,
Praeger and Seress, our main theorem gives a complete classification of all
finite extremely primitive groups, up to finitely many affine exceptions (and
it is conjectured that there are no exceptions). Along the way, we also
establish several new results on base sizes for primitive actions of
exceptional groups, which may be of independent interest.
