Radical Radicals

Algebra Level 3

How many positive $x$ satisfy

$\Large \sqrt{x} + \sqrt{\sqrt{x}} + \sqrt{\sqrt{\sqrt{x}}} + \ldots + \underbrace{\sqrt{\sqrt{\sqrt{\cdots\sqrt{x}}}}}_{n \text{ square roots}}=1$

for a given positive integer $n?$

Bonus: What sorts of bounds can you put on these solution(s) in terms of $n?$