For example, it seems like there "should" be an infinite dimensional hole (or perhaps many) in $S^1 \times S^1 \times \ldots$. (Or perhaps none...) Is there an invariant that would count it? What sort of "interesting" spaces motivate the study of such invariants? Non-finite topological vector spaces minus a point?
Googling infinite dimensional homotopy groups and infinite dimensional homology groups didn't turn up anything that seemed relevant.
A problem with extending homotopy and homology groups to the infinite dimensional case is that the infinite dimensional sphere $S^\infty=\cup_{n} S^n$ (where each $S^n$ includes in $S^{n+1}$ equatorially) is contractible. An easy way to see this is Whitehead's theorem. All homotopy groups vanish, so it must be contractible. Maybe there is some other way to generalize homology or homotopy groups, but the naive way fails.
Here's another example that indicates things are strange when you go to infinite dimensions. The Hilbert cube is the infinite product of closed intervals which you can think of as getting smaller and smaller in diameter. The weird thing is that it is homogeneous. There is a self-homeomorphism taking any point to any point. So, even though it feels like it has a well-defined boundary, it doesn't. In particular taking a point away from it will not change its homotopy type!