Are there any compression algorithms based on PI?
Your suggestion doesn't make much sense, for many reasons. First of all, when trying to compress a large file, say a file of size $16$ bytes, you will have to find a place in the binary expansion of $\pi$ which agrees with your file. Since the file is $128$ bits long, one would expect this place to be around the $2^{128}$th bit. So it would be rather hard to find. This is not only because we have to go far into the expansion, but also since we expect to try $2^{128}$ different locations before finding a hit.

Second, while in some cases your scheme will result in a major compression, this will only happen when a certain string appears comparatively early in the expansion of $\pi$. There is no reason you would ever want to compress that sort of string. In contrast, other compression algorithms try to find structure in the data, and have guarantees that show that if such a structure exists, then they can always exploit it. 

Changing $\pi$ with any other number wouldn't change the picture. The algorithm is too specific, compressing only strings we aren't really interested in; and very inefficient in the compression phase.