Could I, within my lifetime, reach any star I wanted if I went fast enough? Disclamer: I'm not talking about FTL travel here. I'm also not talking about any weird space warping mechanics like wormholes and such.
I've always thought that if a star was 4 light years away, then it would be impossible to reach it with less than 4 years of travel time. Therefore, any star more than 100 light years away would require multiple generations on the ship in order to get there even if they travelled at the speed of light(or close to it). The people who set off on the mission wouldn't be alive when the spaceship arrived at the star.
But the other day I had a realisation that, for anyone travelling close to the speed of light(0.999c), the length between them and the star would contract and they could get there almost instantly(in their reference frame). This also makes sense for someone observing from earth; they would see me take about 100 years to get to this star, but, because I'm going so fast, they would see me barely age. By the time I had got to the star, they would observe me still being about the same age as I was when I set off, even though they are 100 years older. In my reference frame, I would have also barely aged and I would have reached a star that's 100 light-years away in a lot less than a 100 years of travel time.
So is this assessment correct? Could I reach a star that's 100 light years away in my lifetime by going close to the speed of light?
It would be good to get an answer in two parts: In a universe that's not expanding, and in a universe that is expanding.
 A: 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.
A: Yes, assuming you can get close to the speed of light, you can get to the star in your lifetime.
The expansion of the Universe is not important for the length scale of 100 light years.
A: I think your argument is correct. For example, in the Earth reference frame version, you would have aged $L/c\gamma$, where $L$ is the distance from Earth to the star in Earth's frame, and $\gamma=(1-v^2/c^2)^{-1/2}$. Since $v$ can be arbitrarily close to $c$, then $\gamma$ can be arbitrarily large, so your age can be arbitrarily small for fixed $L$. (Of course, it takes increasing huge amounts of energy to increase $v$ as you get close to $c$, which will kill any attempt to do this in practice, but I take it you are ignoring this constraint.)
In terms of an expanding universe, what you would want to do is solve the geodesic equation for a massive particle in an FLRW spacetime, and then compute your proper time to travel on a path that led from Earth to the star. There are definitely cases where you could reach the star: if the time it took you to reach the star in the Earth frame (which is also the cosmic rest frame) is less that the Hubble time (the timescale over which $L$ changes significantly). There are also definitely cases where you could not reach the star: in an accelerating Universe like the one we find ourselves in now, if the star was outside of our cosmological horizon, you can never reach the star no matter how fast you travel. In between those extremes, the answer will in general depend on $L$ and the expansion history of the Universe.
Essentially, the expansion of the Universe will increase the amount by which you will age to reach the star compared to what you would get in a non-expanding Universe, for fixed $v$. If it's possible to reach the star in a finite amount of time, then you can make that age as small as you want by cranking up your velocity. However, depending on the expansion history, there can be some stars where you cannot reach them with any amount of time. These would be stars where light could not travel from Earth to the star.
A: What you've hit upon is that, while your speed $c\beta=c\tanh w$ with $w$ your rapidity is bounded, your proper velocity $c\beta\gamma=c\sinh w$ is not.
Unfortunately, even if you can get all the kinetic energy you want (let's not ask how), you can only safely accelerate so fast, which constrains how far you can travel. Say you accelerate at $g$ for artificial gravity, decelerating in the second half of the journey to arrive at rest: this result implies an $80$-year life can afford about $e^{40}\sim10^{17}$ light years, a few orders of magnitude more than the current radius of our Hubble zone, but then its expansion is accelerating.
How much does the Universe's long-term expansion spoil your plans? That depends. For example, a "big rip" hits an infinite scale factor in a finite time. Less dramatically, suppose dark energy eventually causes an exponential expansion, as in a de Sitter universe. Well, the above scenario would cover exponential ground in a static universe, so it depends on which exponential growth is faster.
A: There is something interesting that I think none of the previous answers mentioned yet: if two events are causally related, an observer can go from one to the other in an arbitrarily small proper time.
Let me phrase this differently. Let $p$ and $q$ be at two events in spacetime such that $q$ is in the causal future of $p$, i.e., there is at least one causal curve from $p$ to $q$. Then it is possible for an observer to start at $p$ and end at $q$ with their clock ticking as few times as they want.
Intuitively, the trick is to make a "zigzag" through spacetime, always keeping your speed close to the speed of light, as depicted in the picture. If you just follow a geodesic from a point to the other, you will maximize proper time. However, when you move close to the speed of light, you can make your proper time as small as desired, as long as you move fast enough. The straight lines in the zigzag are much shorter in terms of proper time than a straight path (i.e. a geodesic) from one point to the other would be. This is a bit counterintuitive at first, but remember that spacetime geometry is Lorentzian, not Euclidean.
Notice this comment is valid in any spacetime. Notice also that it is based on proper time. It means that you can go to Andromeda arbitrarily fast according to your clock. However, when you come back, at least 5 million years (twice the distance from here to Andromeda in light-years) will have passed on Earth.

