Rich pointed me to ( http://arxiv.org/abs/1110.6284 , 37kb) whose abstract ends: "An interesting by-product of our analysis is the evaluation $$_2F_1(1/24,7/24,5/6, -\frac{2^{10}\cdot3^3\cdot5}{11^4})=\sqrt6 \sqrt[6]{\frac{11}{5^5}} $$ and other similar identities." Note the argument > 9.4. Empirically, the two 2F1s produced by the 1/z transformation satisfy Hypergeometric2F1[1/24, 5/24, 3/ 4, -(14641/138240)] == ((45 + 26 Sqrt[3])^(1/3) Gamma[7/12] Gamma[3/4])/ (2^(1/6) 3^(1/24) 5^(19/24) Sqrt[\[Pi]] Gamma[5/6]) and Hypergeometric2F1[7/24, 11/24, 5/4, -(14641/138240)] == (2 3^(17/24) (2 (-45 + 26 Sqrt[3]))^(1/3) Sqrt[\[Pi]] Gamma[1/12])/ (11 5^(13/24) Gamma[3/4] Gamma[5/6]) which together imply the "denesting" -(3 (-45 + 26 Sqrt[3]))^(1/3) + (3 (45 + 26 Sqrt[3]))^(1/3) == 6 --rwg