  We study the secular gravitational dynamics of quadruple systems consisting
of a hierarchical triple system orbited by a fourth body. These systems can be
decomposed into three binary systems with increasing semimajor axes, binaries
A, B and C. The Hamiltonian of the system is expanded in ratios of the three
binary separations, and orbit-averaged. Subsequently, we numerically solve the
equations of motion. We study highly hierarchical systems that are well
described by the lowest-order terms in the Hamiltonian. We find that the
qualitative behaviour is determined by the ratio $\mathcal{R}_0$ of the initial
Kozai-Lidov (KL) time-scales of the binary pairs AB and BC. If
$\mathcal{R}_0\ll 1$, binaries AB remain coplanar if this is initially the
case, and KL eccentricity oscillations in binary B are efficiently quenched. If
$\mathcal{R}_0\gg 1$, binaries AB become inclined, even if initially coplanar.
However, there are no induced KL eccentricity oscillations in binary A. Lastly,
if $\mathcal{R}_0\sim 1$, complex KL eccentricity oscillations can occur in
binary A that are coupled with the KL eccentricity oscillations in B. Even if
binaries A and B are initially coplanar, the induced inclination can result in
very high eccentricity oscillations in binary A. These extreme eccentricities
could have significant implications for strong interactions such as tidal
interactions, gravitational wave dissipation, and collisions and mergers of
stars and compact objects. As an example, we apply our results to a planet+moon
system orbiting a central star, which in turn is orbited by a distant and
inclined stellar companion or planet, and to observed stellar quadruples.
