Can light reach far away galaxies in an expanding universe? I've read that, since the universe is expanding at an increasing rate, light that leaves our galaxy now will never reach far away galaxies. That even though a galaxy is moving away from us slower than the speed of light right now, before the light reaches there from our perspective, the increasing expansion will cause that far away galaxy to be moving away from us faster than the speed of light, and the light will never reach it.
However, it's also my understanding that light doesn't "experience" time (I can't think of a better way to phrase that). So from light's "perspective", it's emitted and absorbed at the same time. So the expansion of the universe that happens during the light's transit from our perspective shouldn't affect it.
I'm obviously misunderstanding something, since the light can't both get there and not get there.
I'm not a physics student, I only have a layman understanding and I'm trying to piece things together. Is the answer simply that the first part of my question is wrong; that the light will in fact reach the other galaxy if that galaxy isn't currently moving away from us faster than the speed of light?
Edit: I'm trying to figure out how it works when a photon leaves our galaxy and heads towards another galaxy, where the two galaxies are moving apart slower than c when the photon leaves, but the expansion of the universe causes the two galaxies to be moving apart faster than c at some point during the photon's transit, from our perspective.
As far as I understand it, a photon only has a transit time when viewed from a relativistic perspective.
 A: Here's an example in layman's terms. It's not 100% accurate, but it should give you a good idea of how it could be.
Imagine a balloon that has only been slightly inflated. Now draw two dots on the balloon. An ant at one dot starts walking towards the other at a constant speed. However, at the same time we start inflating the balloon. Assuming our balloon can keep expanding forever and never pops even as the ant moves towards the second dot that dot is getting farther away because of the balloon's expansion. If we inflate it fast enough the ant will never reach the dot...
Note that in this example the dots aren't even moving on the balloon, yet the expansion could prevent the ant from ever reaching the second dot.
A: '..the increasing expansion will cause that far away galaxy to be moving away from us faster than the speed of light, and the light will never reach it."
This sentence is totally correct. "far away galaxies" move faster than the speed of the light with respect to us. This is not in violation of relativity! General relativity says that nothing can travel faster than light LOCALLY. Locally means  in a very small region of space which is practically flat.
"light doesn't "experience" time". this sentence is not correct. The notion of "from light's "perspective"" is meaningless because your coordinate system can never travel at the speed of the light. In another word, there is no coordinate system in which light is at rest.
From a point of view of a particle which is moving at a speed close to the speed of light, the universe is still expanding and some galaxies are still out of reach.
A: This cannot happen as you describe in our standard $\Lambda$CDM cosmology. That is to say, the answer to

Is the answer simply that the first part of my question is wrong; that the light will in fact reach the other galaxy if that galaxy isn't currently moving away from us faster than the speed of light?

is "Yes". This doesn't mean Aslum's analogy is inaccurate; it could happen in principle in a different universe with a higher cosmological curvature or exotic matter with increasing energy density, just not in ours as we currently understand it.
There is certainly an upper bound to how far from us (measured according to the current notion of distance) a signal we send now will ever reach-- it's about $17$ billion lightyears-- but everything outside of this range is already receding faster than $c$. What's more, there are even points inside this range, i.e. that we can eventually get a signal to, that are currently receding faster than $c$.
The reason it can't happen is that the Hubble parameter is strictly decreasing (ignoring cosmological curvature), though it is leveling off. The distance between two (co-moving) fixed points in the FLRW spacetime is given by
$$D(t) = a(t) D_0 ,$$
where $D_0$ is the current (or co-moving) distance and $a(t)$ is the scale factor. The recessional velocity between these points is then
$$v_r = \dot{D}(t) = \dot{a}(t) D_0 = H(t) D(t),$$
where $H := \frac{\dot{a}}{a}$ is the Hubble parameter.
Now, if we emit a light signal toward a destination point $P$ at current distance $D_0$ as above with recessional velocity $v_r$ currently less than $c$, then after an infinitesimal time, the light will be closer to $P$ than $D_0$ because $v_r < c$. Similarly, as long as $P$ is receding from the position of the light at a velocity $v_r < c$ (it may well eventually recede from us much faster than $c$) the light will get closer to $P$.
If we define $R(t)$ to be the remaining distance to $P$ the light has to travel, then this says that $R(t)$ is strictly decreasing so long as $H(t)R(t) < c$. Now my former point comes up: $H(t)$ is decreasing as well, directly from the Friedmann equations! So if $H(t)R(t)$ is ever less than $c$ (as we've assumed it is initially), then both $H$ and $R$ are decreasing, so $H(t)R(t)$ remains less than $c$. Hence $R$ always decreases, and it will necessarily reach zero.

Addendum Regarding the "perspective" of light: this is a heuristic tossed around in introductory special relativity when discussing time dilation, but there's not much meaning to it. It should certainly not be taken to indicate light actually does anything instantaneously-- it only means that there is no meaningful notion of elapsed time intrinsic to the path light travels. This is distinguished from the case of the timelike paths obserservers can take, which have associated to them their proper time.
It also does not mean that one cannot cook up a reasonable (though highly coordinate-dependent) notion of elapsed time between two points in a spacetime (that may or may not happen to be connected by some light ray). The FLRW spacetime central to cosmology has sufficient symmetry that there's a very physically and geometrically natural choice of global coordinates that induce a notion of the "time" at which each point in spacetime occurs. This is the notion of time virtually universally discussed in the context of cosmology, especially in sources directed at laypeople. This time between two points is independent of any path connecting them, even if it is a path followed by light.
A: In this situation I would prefer to say that light is not affected by time rather than light does not experience time. So, yes light can travel at c in a vacuum and never reach a very distant galaxy where space between them is expanding. > c.
