Oh lor...
The problem is that people are, quite naturally, tied down to what they can experience or easily imagine. So if someone says that the Universe is expanding then most people might think of a balloon (and indeed often that's used as an analogy), that certainly does expand "into something". This is not really the case (or at least, it doesn't have to be). Instead the Universe creates its own spacetime as it expands, and the net result is that objects can be measured as further apart than they were previously.
There is no perfect analogy to what is going on. The best I can do is return to the 2D humans on earth. Being two-dimensional, they need two dimensions to measure their position on their Earth. If they explore it all, they will soon learn that at least one (and in fact both) of these two dimensions is closed -- which is to say that you can describe every point in their Universe using a finite range of numbers -- say, 360 degrees of latitude, and 360 of longitude, and Every point on earth will be covered. In their world, then, someone sitting at 365 degrees latitude, measured from a starting point, would be on top of someone at 5 degrees latitude. So what is beyond 360 degrees? Nothing. Literally nothing. Again, it doesn't make any real sense to ask the question -- all you can do is start describing the same space again with a new set of numbers, but this set is redundant.
This is the kind of situation we are talking about. The closure of space-time in the Universe would imply that you eventually stop needing more numbers to describe your position in the Universe, because you just end up redescribing the same point you were at before. What, then, is beyond? It isn't even a valid question.
The only way it might make sense is if you found -- as our Earth-people might -- that there is some entirely new dimension, extending beyond the confines of the observable Universe. In this analogy, that direction would be up, as opposed to North and East.
Asking what is "Beyond" the Universe, then, only makes sense if either:
i) the dimensions that describe the Universe are bigger than the Universe itself (ie you run out of Universe before you run out of numbers in a certain direction)
ii) there are more dimensions than just three spatial and one temporal direction.
The first is actually true -- so far as we know, at least -- with respect to time. Although the Universe has an apparent beginning, it doesn't necessarily have an end, and may continue to exist forever. We'll have to wait a near eternity to find out though.