can you have your pi and be in your pi too?

Scott reviews Max Tegmark's book about the many different multiverses and, as you would expect, a lot of deep thinking is happening in the comment thread.
Btw it is very likely that the best possible finite description of our world (e.g. all text ever written on wikipedia + arxiv + ...) and not just Scott's blog post, appears infinitely often in the decimal expansion of pi. The same is true for any randomly picked real number.
However, since there are infinitely many real numbers, it is also reasonable to expect that there are many which do *not* contain it even once (e.g. all the natural numbers).
A question for my smart readers: What fraction of the real numbers does not contain such a description of our world?