Here’s a totally impractical but fun back-of-the-envelope calculation from Bob Martin.
Suppose you have a space ship that could accelerate at 1 g for as long as you like. Inside the ship you would feel the same gravity as on earth. You could travel wherever you like by accelerating at 1 g for the first half of the flight then reversing acceleration for the second half of the flight. This approach could take you to Mars in three days.
If you could accelerate at 1 g for a year you could reach the speed of light, and travel half a light year. So you could reverse your acceleration and reach a destination a light year away in two years. But this ignores relativity. Once you’re traveling at near the speed of light, time practically stops for you, so you could keep going as far as you like without taking any more time from your perspective. So you could travel anywhere in the universe in two years!
Of course there are a few problems. We have no way to sustain such acceleration. Or to build a ship that could sustain an impact with a spec of dust when traveling at relativistic speed. And the calculation ignores relativity until it throws it in at the end. Still, it’s fun to think about.
Update: Dan Piponi gives a calculation on G+ that addresses the last of the problems I mentioned above, sticking relativity on to the end of a classical calculation. He does a proper relativistic calculation from the beginning.
If you take the radius of the observable universe to be 45 billion light years, then I think you need about 12.5 g to get anywhere in it in 2 years. (Both those quantities as measured in the frame of reference of the traveler.)
If you travel at constant acceleration a for time t then the distance covered is c^2/a (cosh(a t/c) – 1) (Note that gives the usual a t^2/2 for small t.)