Let $_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;c)$ denote the generalized hypergeometric function. Let $A \subset \mathbb R$ be the set of all values of $\ _pF_q(\cdot)$ at rational points $a_i,b_j,c\in \mathbb Q$. Since $A$ and its algebraic closure $\overline A$ are countable, there are obviously many constants $\alpha \notin \overline A$.

**Question:** Are there any natural constants $\alpha$, as in this list, such that $\alpha \notin \overline A$?

I realize this question is somewhat informal, and in fact I can construct an explicit such constant with rapidly increasing sequences of zeros separated by 1's. I am curious if there is a "nice" such constant for which this is provably (or conjecturally) known. Below is a short list with many "nice" values in $A$.

$_1F_0(1/2;1/2) = \sqrt{2}$

$_0F_1(1/2;1/4) = cos(1)$

$_2F_1(1/2,1;3/2;-1) = \pi/4$

$_2F_1(1,1;2;-1) = \log 2$

$_3F_2(1/2,1/2,1/2;1,1;1) = \pi/\Gamma(3/4)^4$

$_3F_2(1/2,1/2,1/2;1,3/2;1) = 4C/\pi$, where $C$ is the Catalan constant.

$_4F_3(1/2,1/2,1/2,1/2;3/2,3/2,3/2;1) = 7\zeta(3)/8$