Nope. The speed limit applies to the motion of an object as seen by
another which it is passing by. It applies under all circumstances that
we know of.
You are asking about something else- the speed assigned to the
object by some remote viewer back where the object was coming from. It
would seem like very strong gravity could make the object speed up to
greater than c. Your sense that something odd happens is right.
Gravity, it turns out, is not best described as a force within Euclid's
space but rather as an effect on the geometry of spacetime itself. A
very massive compact object creates a black hole, a region of spacetime
from which no objects can depart. That's pretty weird, but it's not the
same as beating the speed limit in the ordinary sense.
General relativity starts with the idea that small bits of
spacetime act as ordinary, "flat" spacetime where the effects of
gravity are small. It turns out that there is no set of flat Lorentzian
coordinates which works over large pieces of spacetime when gravity is
allowed. So the speed limit of special relativity, when written in GR
context, is that nothing can fly past something else at a speed which
exceeds c. But if you are comparing relative speeds of objects that are
separated by large distances or large times, the gravitational warping
of space can make your statements about their relative speeds
The universe is expanding, and if it is uniform on very large
scales (and it seems to be), then galaxies and stars that are beyond
our horizon are indeed moving away from us at a speed faster than the
speed of light in our coordinates. But nothing about them lets them
move faster than the speed of light in their local coordinates. A
description of this is provided in this article at Scientific American
which is fortunately available online at least for the moment. It's in the March 2005 Issue.
(republished on 07/23/06)