How long would it take to reach the edge of the reachable universe? How long would it take to reach the current edge of the reachable portion of the universe, with the following bounds in mind:


We figure out a way build a space ship to provide a constant 1 G acceleration for an extended period of time.
We accelerate until we are halfway there, then decelerate past this point (so we stop when we get there.
Our target is about 15 billion lightyears away.  As per http://curious.astro.cornell.edu/about-us/104-the-universe/cosmology-and-the-big-bang/expansion-of-the-universe/616-is-the-universe-expanding-faster-than-the-speed-of-light-intermediate galaxies past 4,740 megaparsecs are no longer reachable. Using this site http://www.convertunits.com/from/megaparsecs/to/light+years to convert megaparsecs to lightyears means this "unreachable" point is currently 15,459,812,396.5002 lightyears away.  Let's simplify to 15 billion lightyears, which is still supposed to be reachable.
We somehow work out all the details of life support, collision with particles at relativistic speeds, etc...


I would like to know the time the traveler experiences, and the time that everyone else (e.g. those on earth) experience.
Jonathan's answer is essentially correct, but as Rob Jeffries comments, he doesn't take into account that the Universe is expanding during the journey.

The edge of the observable Universe is 47 billion lightyears (Gly) away. Even if you are a lightbeam, you cannot reach that point. The farthest you can go if departing today is roughly 5 Gpc, or 17 Gly, but this journey would of course take infinitly long (or else it wouldn't be "the farthest you can go"). This distance is probably what the linked article is referring to (I didn't read the article; it's very, very long).

So, in order for the answer to be any fun, you have to freeze the Universe, using magic, which is what Jonathan's calculator is doing. Here I'll just provide the analytical solution: In that case, the proper time $\tau$ (i.e. the time experienced by the traveler) to reach a distance $x$ when traveling at a constant acceleration $a$ is
$$
\tau = \frac{c}{a} \cosh^{-1} \left( \frac{ax}{c^2} +1 \right),
$$
where $c$ is the speed of light. If you wish to decelerate after having reached halfway, you just divide $x$ by $2$ and multiply the result by $2$.

If you plug in the $x=15\,\mathrm{Gly}$ you request, you get roughly 45 years. To get to the edge of the Universe at 47 Gly actually only takes a few years more. The reason for this is simply that traveling at 1G gets you to (almost) the speed of light in only a couple of years, and hence you experience (almost) no time, no matter how far you go.

The time experienced for the Earthlings for the traveler at constant acceleration is given by
$$
t(\tau) = \frac{c}{a} \sinh \left( \frac{a \tau}{c} \right),
$$
which works out to 15 Gyr for the 15 Gly, and 47 Gyr for the observable Universe. The reason is simply that the traveler, from the point of view of the Earthlings, extremely fast reaches a speed which is almost the speed of light.