How can we know there is more than just the observable universe If all we can observe are things inside the observable universe, how do we know that anything even exists outside this boundary? I can see four ways of solving this problem.
1) We wait a while, the observable universe should get 'larger', so we should be able to observe more. I don't think this is practical though, since telescopes have only existed for a hundred years or so, whereas the age of the universe is many degrees larger. Also, galaxies are expanding outwards, so they will probably escape the observable universe horizon before we can observe them.
2) We go to another planet, and the 'centre' of the observable universe there is different, so in effect we can 'see' further then we could before. This is also impractical, we can't travel large enough distances for this effect to even have an inkling of an effect.
3) No way of knowing. All the matter that is within the observable universe could be all that there is, and we would have no way of knowing if anything even exists outside. 
4) Theoretical calculations. Well scientists seem to be able to calculate (from redshifts?) the amount of dark matter, dark energy, and matter in the universe. That would mean that we would gain an understanding of what SHOULD be outside our observable universe (though we may never see it). This seems the most logical, but I have no idea how these calculations are carried out, any insight would be appreciated.
 A: As it happens this issue has just been mentioned in the Science Fiction Stack Exchange. The simple answer is that we cannot know that there is anything outside the observable universe. The best we can say is that it seems likely.
We approximate the universe using the a spacetime geometry called the FLRW metric. This is based on the assumption that the universe is the same everywhere - technically that it is homogeneous and isotropic. If the universe is the same everywhere then obviously it's the same beyond the bits we can see, but remember that this is just an assumption.
It's possible to construct a universe that looks like an FLRW universe locally but contains no matter outside some boundary. This metric is called the Oppenheimer-Snyder metric and was devised as an approximate description of a collapsing star forming a black hole. However we can reverse the time direction and the metric then describes matter emerging from a white hole. As long as the boundary is farther away than the edge of the observable universe we would not be able to tell the difference between an Oppenheimer-Snyder universe and an FLRW universe.
However the Oppenheimer-Snyder construction is a rather artifical one. It is created by welding together a patch of spacetime described by the FLRW metric and a patch described by the black hole (Schwarzschild) spacetime. While it's technically possible for this geometry to describe our universe it would take a deity with a rather peculiar sense of humour to arrange the universe in this way. The FLRW universe seems much less contrived and therefore more likely.
A: What we do know is that the universe is expanding at an accelerating rate. (you've been incorrect in formulating that the observable universe gets larger with time, the observable universe actually gets smaller, with more and more galaxies and stars getting red-shifted to the point of invisibility.) So to claim that there is nothing more than the observable universe is incorrect because since our observable universes magnitude (apparent) has been decreasing, more and more galaxies are out of sight, so we can technically prove that there are galaxies which were once visible, and currently are not, and hence there exist objects outside the "observable" universe.
I hope this has been clear enough.
Edit:

The radius of the observable region would remain somewhat same, but the objects which are to be "observed" move out of it due to space expansion, and hence the observable objects reduce, hence causing a decrease in the size of the observable universe, if we agree on the definition of the word.
