Space-time cannot be defined only by its 3 dimensional volume. The 4 dimensional space-time continuum is continuously expanding, but anything that was somehow located outside of the space-time continuum would be undefined and unexplainable by our laws of physics. It's easier to run the film backward to understand it. As the universe shrinks, the volume of space-time gets smaller and smaller until it becomes a singularity. All physics breaks down at that point, time, dimension, position, spin energy all become meaningless and undefined, since the laws of the universe are unified under those conditions. In a universe with a absolute size of 0, how do you define motion? At the moment of the Big Bang, and ever since, space-time has expanded like a giant loaf of bread, dragging all the matter and energy in the universe along with it. Whether the undefined space is being displaced by our space-time, or if space-time has its own boundary is irrelevant. Anything "outside" of our universe would almost certainly not be governed by the physical laws that exist in space-time. If the edge of space-time was interacting with whatever medium lies beyond its defined limits, wouldn't we expect to see some kind of quantum mechanical activity along the interacting edges?
The universe may be able to expand infinitely, perhaps leading to the Big Rip, where the universe will shear apart, perhaps ultimately ripping space-time itself apart...who knows? We can only perceive things that are within space-time and that we can interact with. Outside of spacetime, our understanding of physics is likely not applicable, and we cannot perceive what might lie beyond, except by any effects it might have upon space-time itself, which appears to be none. Anything outside of space-time, unless it coincidentally has exactly the same properties as space-time itself (which is unlikely) is undefinable by our physics. I believe that if you could somehow warp to the edge of space-time, and then push beyond its 'frontier' you would simply drag/stretch space-time out with you along that vector. The matter and energy of you and your spaceship are defined by the rules of space-time, so even though space-time can be stretched and even broken (black holes, for instance) the mass-energy remains in our universe, albeit in a strange form...