What is the expanding universe expanding into?

So, let's bring physics into this. According to Einstein's General Relativity, the universe belongs to a particular type of geometry called a Riemann manifold (actually pseudo-Riemann since the Einstein view of the universe does not have the positive definite metric of Riemann's original work). What this means is that space-time has an intrinsic metric (which is the solution to his field equations) that does not require the universe to be embedded in anything. In other words, the universe can be described by it's intrinsic geometry without reference to anything external. In Einstein's equations, there is NO reference to anything outside the Riemann manifold.

So, the bottom line is the universe does not need to be expanding into anything - as Morningfox stated and for some erroneous reason got two thumbs down for it. However, it does not eliminate the possibility that it could be expanding into some sort of embedding space - the Riemann manifold does not rule that out. One could appeal to Occam's razor here but that wouldn't be very scientific.
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It's expanding into the universe and if the universe is contained within something else it's expanding into that. At the end of this universe or the Russian doll like arrangement of universes there is infinity. Infinity is inescapable. All roads lead to forever.
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It's expanding into the void or the multiverse, if you like. What the Big Bang did was that it organized the otherwise disorganized dimensions of the multiverse into a universe with a purpose. Some theories suggest that there may be as many as 10 or 11 dimensions. The Big Bang took 4 out of those dimensions and inflated 3 of them slowly (space), and one of them very fast (time).