Mathematicians David Bailey of the Ames Research Center, Peter Borwein of Simon Fraser University and Simon Plouffe of the Centre for Experimental and Constructive Mathematics (an organization located on the campus of Simon Fraser) back in 1995 developed an algorithm for the computation of the nth digit of π,

π = sum<n=0 .. Infinity> ( T * (1/16)^n)
where
T = 4/(8n+1) - 2/(8n+4) - 1/(8n+5) - 1/(8n+6)

The kicker here though is that a single digit of the transcendent number is extracted, and as the (1/16)^n term indicates, that digit is in hexadecimal. Other algorithms must be used to convert the partial series of hexadecimal digits into decimal, or base 10.

The full announcement of the discovery of this algorithm, along with the results of their calculating the ten billionth digit of π in base 10 (which is a '9'!) can be found at http://www.mathsoft.com/asolve/plouffe/announce.txt. A description of how the algorithm works for a non-technical audience is available at http://www.mathsoft.com/asolve/plouffe/elmntary.txt.