  An unusual set of orbits about extreme Kerr black holes resides at the
Boyer-Lindquist radius $r = M$, the coordinate of the hole's event horizon.
These ``horizon skimming'' orbits have the property that their angular momentum
$L_z$ {\it increases} with inclination angle, opposite to the familiar behavior
one encounters at larger radius. In this paper, I show that this behavior is
characteristic of a larger family of orbits, the ``nearly horizon skimming''
(NHS) orbits. NHS orbits exist in the very strong field of any black hole with
spin $a\agt 0.952412M$. Their unusual behavior is due to the locking of
particle motion near the event horizon to the hole's spin, and is therefore a
signature of the Kerr metric's extreme strong field. An observational hallmark
of NHS orbits is that a small body spiraling into a Kerr black hole due to
gravitational-wave emission will be driven into orbits of progressively smaller
inclination angle, toward the equator. This is in contrast to the ``normal''
behavior. For circular orbits, the change in inclination is very small, and
unlikely to be of observational importance. I argue that the change in
inclination may be considerably larger when one considers the evolution of
inclined eccentric orbits. If this proves correct, then the gravitational waves
produced by evolution through the NHS regime may constitute a very interesting
and important probe of the strong-field nature of rotating black holes.
