Is there a general identity for the infinite radicals; $\sqrt{n^{0}+\sqrt{n^{1}+\sqrt{n^{2}+\sqrt{n^{3}+...}}}}$

You are studying the limit of nested radicals of the form

$$a_{k} = \sqrt {x^0 + \sqrt{x^1+\cdots +\sqrt{x^k}}}$$

There is a theorem by Herschfeld (On infinite radicals, 1935) that shows $a_k$ converges if and only if the sequence

$$F_k := (x^k)^{1/2^k} = x^{k/2^k}$$

is bounded. This happens for every $x\in \mathbb{N}$.

In the article by Herschfeld there are several results on nested radicals (including the case for $x=n^2$ which comprises your request $x=4$) of this form but also of the "left-hand" form

$$u_k = \sqrt{a_k + \sqrt{a_{k-1}+\ldots +\sqrt{a_0}}}$$

of which the author is able to calculate the general limit in terms of the limit of sequence $\{a_k\}$.