There are at least two ideas involved.
First is that the expansion of the universe is not linear. While the Big Bang happened around 14B years ago, that does not mean that 13B years ago, the Universe is 1/14th of its present size. Current theory suggests that a large portion of the cosmological inflation (where the Universe increased by 26 or more orders of magnitude in linear dimensions) happened within much, much less than a second after the Big Bang. And as another example, the current theory estimates that at the time that the cosmic microwave background was emitted (which was about 0.5 million years after the birth of the Universe, placing it about 1/30000 the current age of the Universe), the universe is already about 1/1000 its current size (in length).
Second is that the apparent recession of far away objects from us is not so much objects flying apart from each other. Rather, it is space being added in between objects. Imagine you being the photon, and two turtles (moving slower than you) being the galaxies. Put turtle one in the first carriage of a train, and put turtle two on the 10th carriage of a train. And you start walking. Say it takes you 1 minute to traverse a carriage, and it take the turtles 10 minutes. Then in the case where the turtles walk away from each other, it will take you a bit under 12 mintues to get from the first turtle to the second (you walk 10 minutes to the tenth train, and the turtle has gotten to the 11th. You walk another minute to the 11th train. The turtle is just a few steps in front of you.)
But that's not how the universe expands. The expansion of the universe is more like the following: suppose every 6 minutes, all the carriages decouple, and between each pair of the original carriages plops one more car! So you walk for 6 minutes (having traversed 6 cars), and you look up, and see that the second turtle is 8 cars in front of you (and the first turtle is 12 cars behind). And you walke another 6 minutes. Plop comes the extra cars, and now you are 4 cars from the second turtle and 36 cars from the turtle behind. And finally after another 4 mintues you catch up to the second turtle.
From the point of view of the second turtle though, you would have travelled from a turtle that is now 40 cars away from him, while taking only 16 minutes! This ties back into the funny idea that light emitted from an object 13B lightyear away can take quite a bit less than 13B years to get here, due to the inflationary Universe.
This is why cosmologists and astronomers use red-shift to measure distance, because there is no reasonable intrinsic notion of distance that is free from ambiguity: should distance be described by how far away the turtles are when you started walking? or when you finished walking? or the number of carriages you (the photon) have traversed? Instead of that, they measure it using red-shifts, which can roughly fit into this turtle-you framework as how flushed your cheek is from all that walking when you reached turtle number two. Based on the redness of your cheeks, the turtles can calculate how much you exerted yourself, and thus for how long you've been traveling, and using known rules of the addition of new cars (the value of Hubble constant), the turtles can estimate the distances to other turtles. :-)
(I'm going to skip discussion of standard turtles, which are turtles from which you will always depart well rested and not flushed, nor how the turtle simiano-ferroequinologists found out about their rates of locomotive expansion.)
Your question can be translated into "if right now we would send a powerful omnidirectional light pulse from earth into space, would there be galaxies that never see this light pulse?"
The answer is "yes". Due to the accelerated expansion of the universe, as described by the lambda-CDM model, only galaxies currently less than about 16 billion light years (the difference between the cosmological event horizon and the current distance to the particle horizon) away from us will at some time observe the light pulse.
A nice visual representation of this can be found in figure 1 of this publication.
Best Answer
No, you cannot. The do-not-attain speed is 99.99999999999999999998% of the speed of light, at which point your interactions with the cosmic background radiation are blue-shifted to the proton-degneration resonance energy. This will erode your spacecraft and you to something rather unrecognizable over a distance of 160 million light years. This limit is known as the Greisen-Zatsepin-Kuzmin limit and also known as the zevatron limit since a proton at that speed has about 1 ZeV energy and energy rather than velocity is what's observed in catching a cosmic ray.
See the Greisen-Zatsepin-Kuzmin limit.
Thus, your maximum Lorentz factor is $5 \cdot 10^{10}$ and therefore your maximum attainable distance is $5 \cdot 10^{12}$ light years.
While this exceeds the radius of the current observable universe ($4 \cdot 10^{10}$ light years), after taking into account the expansion of the universe, there are stars within our future light cone that you cannot reach.