This infinite product is basically a mathematical constant called Q in digital tree searching. There is some helpful information on http://mathworld.wolfram.com/InfiniteProduct.html and http://mathworld.wolfram.com/TreeSearching.html.
I am actually trying to calculate a probability and hitting upon the infinite product:
\prod_{j=1}^{\infty}\frac{2^{j}-1}{2^{j}}
Any idea what this might be (it's about 0.28, but I want the formula).