This question while asking for something different brings up an intriguing problem: If you can find $x$ such that $x\equiv R^x\pmod p$ then you can break DSA. Now I thought that one might be able to relate this to the discrete logarithm problem somehow, but I failed to find the relation and so I'm asking this (somewhat more general) question.
Now my question is:
Let $p$ be an arbitrary prime, does there exist an algorithm faster than $\mathcal O(p)$ which, given $p$ and an arbitrary $R\in\mathbb Z$, finds an $x\in\{y\in\mathbb Z\mid 0\leq y<p\}$ such that $x\equiv R^x\pmod p$ if such an $x$ exists or $\perp$ otherwise?