On the distance between factorials and repunits

We show that if $n\ge n_0$, $b\ge 2$ are integers, $p\ge 7$ is prime and $n!-(b^p-1)\mathrm{/}(b-1)\ge 0$, then $n!-(b^p-1)\mathrm{/}(b-1)\ge 0.5\log \log n\mathrm{/}\log \log \log n$. Further results are obtained, in particular for the case $n!-(b^p-1)\mathrm{/}(b-1)<0$.