Is there a general form for infinite nested radicals of degreen n?
Solution 1:
If $n=5$, $a=1$, the Galois group of $x^5-x-1$ is $S_5$. So a root cannot be expressed in radicals starting with rational numbers by the Galois theorem, so no algebraic solution is possible if $a=1,n=5$.
For arbitrary $a$, $n=5$ and more general solutions see this question.