In another thread
I calculated the positions of a super 17x17x17, which I did with my own approach of handling each piece type one at a time. So I think I'll show it here as well.
For Oskar's 17x17x17, there are 56 groups of centers(ignoring the central center pieces), each of which contain 24 centers. 4 in each set have the same color, and there are 6 colors for them as well. This means there are:
((24!^56)/(4!^336))=4.371*10^868 possible combinations.
For a normal 17x17x17, we must account for the edges and corners.
Corner orientations: (3^7)
Corner permutations: (8!)
Middle edge permutations: (12!/2)
Middle edge orientations: (2^11)
There are 7 groups of wing edges, each containing 24 edges, which gives (24!^7) for their permutations, which are not restricted to even parity.
Multiplying these all together gives:
(8!)*(3^7)*(12!/2)*(2^11)*((24!^56)/(4!^336))*(24!^7)=6.691*10^1054 possible combinations.
(EDIT: fixed an error after Brandon pointed it out)
Are you making a "Why is the Rubik's cube so hard" episode 3? If so, maybe you could use Planck volumes to make a comparison.