No. This is because all hypergeometrics are holonomic, and holonomic functions can only have a finite number of singularities, which themselves can only be of certain types. If the logarithm of all hypergeometrics could be so expressed, then you could have a holonomic function with a $\ln \ln (x)$ singularity, which is not possible.I'll add some references
I find the paper On the non-holonomic character of logarithms, powers, and the nth prime function by Flajolet, Gerhold and Salvy (The Electronic Journal of Combinatorics, 2005, vol. 11) to this laterbe a wonderful compendium of useful tools for disproving holonomicity. Searching through the literature to find these tools is tedious, and so these authors ought to be commended for assembling so many into one pleasant paper.