Expansion of the Universe, will light from some galaxies never reach us? Is it true that the light from some galaxies will never reach us? 
The explanation for that is that the Universe expanding faster than the speed of light. But, if the speed of light is constant in all reference frames, then in the reference frame of a galaxy, light must move at the speed of light and thus it must reach us (?). 
Could you also explain  how is it possible that the Universe is expanding faster than the speed of light if an object can never be observed to move faster than light? (without Doppler effects).
 A: Let me present a slightly different perspective to Luboš, though I'm saying basically the same thing. From our current location we can define an area of space called the future light cone. This is the region of spacetime that is connected to us by motion at less than or equal to the speed of light. If we draw a spacetime diagram then the lightcone looks like:

Anything that is within our future light cone will always stay within our future light cone. As an aside, this is why nothing can ever fall into a black hole in our coordinates because crossing the event horizon would take it outside our light cone. Instead we see the object freeze at the event horizon.
But back to the universe: your argument that:

an object can never be observed to move faster than light

is true for everything in our future light cone, but the bits of the universe that are moving faster than light relative to us, and that we will never see, were never in our (past) light cone. This basically because the Big Bang wasn't an explosion outwards from a single point (as misleadingly shown in most TV documentaries).
However, as Luboš says, if we wait long enough even the most distant galaxies will eventually enter our light cone. Well, probably. This is always true for a decelerating expansion, and is even true for accelerated expansion provided $\dot{a}$ increases more slowly than $a$. See the paper Expanding Confusion for the gory details.
A: I am aware that my answer can sound surprising, too simple to be true, but please take a deep breath before downvoting.The answer has little to do with relativity. 


*

*In SR it is the moving object that gets shorter , but space is stable. In such a universe, even if a body is receding at 2,3,30 c, its light will reach us sometime, and the time is short as it is simply D/C. That is because once the photon is discharged, what the sources does is absolutely irrelevant.

*In a universe where space is not stable but is stretching (FLRW) the situation is different because D is increasing. It might seem obvious then that light, on certain conditions, might never reach us. It is not so: 
It is counter-intuitive, but no matter how fast D is stretching, light will always reach us, and you find a good math explanation here, of course in some cases it will take a very looooong time.
This is not the case here, as the fastest acknowledged rate is about $\pi$ C.
This is what lubos Motl is conceding , only as a codicil:

One should perhaps also point out that in some distant future, any
  galaxy (or the place where it lived before it ran out of energy) is
  ultimately going to be visible from Earth.

For an obscure reason, cosmologist adore to make simple things look complicated
A: I am sorry to say that I can not agree with previous answers.  We believe, but do not know for sure, that light from some galaxies will never reach us.  This has nothing to do with the fact that they are moving away from us at more than the speed of light.  Rather, it is assumed that these galaxies, like us, are not moving relative to the special frame in the universe: the one in which large clusters of galaxies are stationary.  They seem to be moving away from us because the universe is expanding.  And,  if there is "dark energy" (as we currently think) and that dark energy persists for long enough, then the expansion of the universe will accelerate more and more with the passage of time.  It is this acceleration of the expansion of the universe that implies that light from sufficiently distant galaxies will never reach us.  
And, actually, it is not known that light from sufficiently distant galaxies will never reach us.  There are (at least) two possible reasons for this:
a) we are wrong about details of dark energy
b) the universe is not infinite but really is finite: something like (say) the surface of a sphere or donut, closed in on itself.  If this is the case, it maybe we will see all the galaxies.  We will just not see all the repeat images of each galaxy, as you keep going around the donut.
A: The relative speed between two objects is only restricted within the special theory of relativity. These restrictions are only guaranteed to apply in general relativity – the theory of curved space that you need for the Big Bang theory – if the space surrounding the objects is the flat Minkowski spacetime, or at least can be approximated by the flat Minkowski spacetime.
In practice, it means that special relativity is guaranteed to hold locally, in very small regions of spacetime which are always nearly flat if they are small enough. That's why the relative speed of two objects that are just passing by one another can't exceed $c$. Special relativity would also (approximately) apply in much larger regions of the spacetime if those were (nearly) flat – if the Riemann curvature were zero (or small) everywhere.
But if you consider the distant galaxies that are receding very quickly – comparably to the speed of light or faster – away from us due to the expansion of the whole Universe, then the condition of the flatness of the spacetime in between the two galaxies, our and theirs, is explicitly violated. That's why the restriction from special relativity no longer holds.
One should perhaps also point out that in some distant future, any galaxy (or the place where it lived before it ran out of energy) is ultimately going to be visible from Earth. That's because the Universe is getting older and we can therefore see further.
