[stouset]

This kind of nerd-sniped me. I did find a closed-form solution, that relies on an identity mapping the product of successive factorials ("hyperfactorials") to the Barnes G-Function which is related to the Riemann Zeta Function at a level deep past my comprehension. Still, here's[1] the closed form solution which returns the number of bits of entropy able to be generated given some number of coin flips using this approach, assuming the coin is indeed fair. Divide by the number of flips again to get the efficiency.

Unfortunately, Wolfram Alpha isn't able to determine the limit of this function[2], and neither am I. :)

[1] https://www.wolframalpha.com/input?i2d=true&i=evaluate+Divid...

[2] https://www.wolframalpha.com/input?i2d=true&i=Limit%5BDivide...