Nested radicals and n-th roots There are many beautiful infinite radical equations,
some relatively straightforward, some much more subtle:
$$
x = \sqrt{ x \sqrt{ x \sqrt{ x \sqrt{ \cdots } } } }
$$
$$
\sqrt{2} = \sqrt{ 2/2 + \sqrt{ 2/2^2 + \sqrt{ 2/2^4 + + \sqrt{ 2/2^8 + \sqrt{ \cdots}}}}}
$$
$$
3 = \sqrt{1 + 2\sqrt{1 + 3\sqrt{ 1 + 4\sqrt{ \cdots }}}}
$$
But I have seen far fewer analogous equations for $n$-roots.
Here is one:
$$
2 = \sqrt[3]{6 + \sqrt[3]{6 + \sqrt[3]{6 + \sqrt[3]{\cdots}}}}
$$
My question is:

Q. Are there truly "more" beautiful infinite radical equations,
  or is it just our natural gravitation toward the simpler square-root equations that
  leads to collections emphasizing radicals?

I am aware this question is vague, but perhaps some nevertheless have insights.

Here's a nested radical for 4th roots that is not so well known. Given the tetranacci numbers (an analogue of the fibonacci numbers). Let $y$ be the tetranacci constant, or the positive real root of,
$$y^4-y^3-y^2-y-1=0$$
Then,
$$y(3-y) = \sqrt[4]{41-11\sqrt[4]{41-11\sqrt[4]{41-11\sqrt[4]{41-\dots}}}} = 2.06719\dots$$
The family can be found in this MSE post.
