As a demonstration of the non-intuitiveness of measures in high dimensions, I think this is still a decent example, no? I agree that n=5 [sic on my part above] isn't special, but seeing the Gamma function should be cause enough for worrying your intuition. It might be misleading, but I'd hope that it wouldn't beg the question of what's special about n=5, but instead what's wrong with your views about high dimensional space.
I agree with the comments on MathOverflow though.
Check out Bill Thurston's comment on why this is deeply misleading geometrically:
http://mathoverflow.net/questions/53119/volumes-of-n-balls-w...
As several commenters there point out, if you must choose a ratio, a more geometrically natural choice would be the ratio of the unit sphere to its circumscribed cube (which decreases monotonically) or to its inscribed cube (which increases monotonically).