Victor Miller: It turns out that one can find an effective irrationality measure for pi/sqrt(3), which should suffice for Rich's algorithm (I think). Here's the title and abstract:

Irrationality measures of $\log 2$ and $\pi/\sqrt{3}$ Nicolas Brisebarre Source: Experiment. Math. <http://projecteuclid.org/handle/euclid.em>Volume 10, Issue 1 (2001), 35-52. Abstract Using a class of polynomials that generalizes Legendre polynomials, we unify previous works of E. A. Rukhadze, A. K. Dubitskas, M. Hata, D. V. and G. V. Chudnovsky about irrationality measures of $\log 2$ and $\pi/\sqrt{3}$ --well, this will yield an RCS-style algorithm for pi*sqrt(3) which according to eq18 in this compendium by D.H. Bailey http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf is a BBP number. However, I do not see why it yields an RCS-style algorithm for pi itself? Oh, I see, MIller's idea is that you square it and that square must be far from any rational. A large subset of the BBP numbers in Bailey's compendium are "linear forms in logarithms" in Baker's terminology, or polynomials in them, and therefore by the Baker-Wustholz explicit transcendence bound I cited will yield RCS-style min-memory algorithms. However, some of the BBP numbers in the compendium instead are "polylogarithms" like EQs 25, 29, 30, 32, 53, 55... and for them it would seem you are helpless except perhaps in the case of zeta(3) for which irrationality is known and perhaps explicit bounds. [Really, we do not need that our number be far from all rationals, only from rationals with denominator=power of 2.]