Are there BBP formulas for all computable numbers?
You can easily use diagonalization to construct a computable real which is not of this form. Fix some enumeration of all triplets $(b,p,q)$. I will describe a process that generates binary digits of a real number.

The process goes over all triplets $(b,p,q)$ in the fixed order. Suppose that it's time to handle $(b_i,p_i,q_i)$, and that currently we have generated the digits $\beta_1 \dots \beta_n$. The process evaluates the corresponding sum $S(b,p,q)$ to such a precision that the first $n+1$ binary digits can be gleaned for it. It outputs $\beta_{n+1}$ so as not to equal to $S(b,p,q)$.

The argument as stated is not 100% correct, and I'm leaving it to the reader to discover where I cheated and to fix the proof.