Quote by turbo1
JesseM pointed out a third possibility  that the Universe can be locally flat/Euclidean, but have a nontrivial topology that allows it to curve back on itself. The website he links uses a torus as an example, but others use Klein bottles, teacups, manifolds with multiple interconnedctions, etc, etc. Other than the possibility of seeing selfsimilar constructs on opposite sides of the Universe (which has never been done, even in the WMAP data) I am not aware of any testable predictions by which these mathematical curiosities might be falsified. In science, something that cannot be falsified (for instance a statement like "angels are pushing the Universe apart, causing expansion") has no standing. Just because it is mathematically possible to do geometry in a topologically nontrivial frame, that does not make the chance that our Universe has assumed that topology likely, nor even possible.
Using Occam's Razor (apparently flat universe, without assuming a complex nontrivial topology) I believe that you will have to accept a spacially infinite Universe with no center from which it all began, as JesseM stated earlier.

If neither an infinite universe with the simplest topology nor a finite universe with a nontrivial topology make any distinct predictions, how can it possibly be a scientific question which one is actually true? An argument which uses "Occam's razor" is a purely metaphysical one if it is impossible
in principle to test whether your conclusion is correct (unlike, say, the theory that the laws of physics work differently on a single planet in the Andromeda galaxy, an idea which seems very implausible by Occam's razor, but which could in principle be tested directly). Plus, some people might argue that a finite universe is inherently simpler than an infinite one, and is therefore favored by Occam's razor even if it requires a slightly more complicated topology. As it happens, it could actually be possible to find experimental evidence for a finite universe by looking for repeating patterns in the cosmic microwave background radiation (see
this page, or
this one), but this would only work if the radius of the universe is smaller than the maximum distance we can observe.