Find $\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\cdots}}}}$ 
Find the value of $$\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\cdots}}}}$$

I know how to solve when all surds are of the same order, but what if they are different?
Technically, (as some users wanted to know exactly what is to be found), find:
$$\lim_{n\to\infty}\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\cdots+\sqrt[n]{4}}}}} $$

Put $$y=\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+...}}}}\qquad (1)$$ stopping successively at the $n$-th root we have a sequence  strictly increasing and bounded  so the limit $y$ is well defined.
From $(1)$ we can easily get a sequence $\{P_n\}$ of polynomials such that its largest real roots $$\alpha_n=\sqrt{4+(4+(4+(4+….(4+\sqrt[n]4)^{1/n}….)^{1/5})^{1/4})^{1/3}}$$
form a sequence $\{\alpha_n\}$ converging to $y$. In fact we have
$$\begin{cases}P_2(x)=x^2-4\\P_3(x)=(x^2-4)^3-4\\P_4(x)=( (x^2-4)^3-4)^4-4\\.....\\.....\\P_n(x)=(P_{n-1}(x))^n-4\end{cases}$$
We can see the polynomial $P_n$ has degree $n!$ and $P_n(\alpha_{n-1})=-4$. Since $P_n(x)$  grows very quickly, this indicates that $\{\alpha_n\}$  rapidly converges to the limit $y$ because the arc of the graphic of $P_n(x)$ between the points $(\alpha_{n-1},-4)$ and $(\alpha_n,0)$ is almost a vertical line so $\alpha_{n-1}\approx \alpha_n$. This argument is enhanced by the equality 
$$y=2\sqrt{1+\left(\frac{4}{4^3}+\left(\frac{4}{4^{12}}+\left(\frac{4}{4^{60}}+….\left(\frac{4}{4^n}+\sqrt[n]4\right)^{1/n}….\right)^{1/5}\right)^{1/4}\right)^{1/3}}$$ hence the factor of $2$ is equal to $$\sqrt{1+\left(\frac{1}{2^{6-2}}+\left(\frac{1}{2^{24-2}}+\left(\frac{1}{2^{120-2}}+….\left(\frac{1}{2^{n!-2}}+\sqrt[n]4\right)^{1/n}….\right)^{1/5}\right)^{1/4}\right)^{1/3}}$$ or
$$\sqrt{1+\left(\frac{1}{2^{4}}+\left(\frac{1}{2^{22}}+\left(\frac{1}{2^{118}}+….\left(\frac{1}{2^{n!-2}}+\sqrt[n]4\right)^{1/n}….\right)^{1/5}\right)^{1/4}\right)^{1/3}}$$  This factor of $2$ grows very slowly obviously.
It seems to me there is not a closed form for $y$ (It could be an asymptotic one perhaps). Consequently I give here an (justified) approximation taking the above referred real root of, say, $P_6$; we have$$y\approx\sqrt{4+\left(4+\left(4+\left(4+\left(4+\sqrt[6]4\right)^{1/6}\right)^{1/5}\right)^{1/4}\right)^{1/3}}\approx \color{red}{2.40161550315}$$  
