Closed form of $\frac{e^{-\frac{\pi}{5}}}{1+\frac{e^{-\pi}}{1+\frac{e^{-2\pi}}{1+\frac{e^{-3\pi}}{1+\ddots}}}}$ It is well known that
$$\operatorname{R}(-e^{-\pi})=-\cfrac{e^{-\frac{\pi}{5}}}{1-\cfrac{e^{-\pi}}{1+\cfrac{e^{-2\pi}}{1-\cfrac{e^{-3\pi}}{1+\ddots}}}}=\frac{\sqrt{5}-1}{2}-\sqrt{\frac{5-\sqrt{5}}{2}}$$
where $\operatorname{R}$ is the Rogers-Ramanujan continued fraction:
$$\operatorname{R}(q)=\cfrac{q^{\frac{1}{5}}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\ddots}}}},\, q=e^{\pi i\tau}.$$
But I'm interested in $\operatorname{R}(e^{-\pi})$. Numerically, I checked that it agrees with the root of the following octic equation near $x=\frac{1}{2}$ to at least $16$ decimal places:
$$x^8+14x^7+22x^6+22x^5+30x^4-22 x^3+22 x^2-14x+1=0;$$
the root turns out to be equal to
$$\frac{\sqrt{5}-1}{2}\frac{\sqrt[4]{5}+\sqrt{2+\sqrt{5}}}{\sqrt{5}+\sqrt{2+\sqrt{5}}}.$$
So is it true that
$$\frac{e^{-\frac{\pi}{5}}}{1+\cfrac{e^{-\pi}}{1+\cfrac{e^{-2\pi}}{1+\cfrac{e^{-3\pi}}{1+\ddots}}}}=\frac{\sqrt{5}-1}{2}\frac{\sqrt[4]{5}+\sqrt{2+\sqrt{5}}}{\sqrt{5}+\sqrt{2+\sqrt{5}}}?$$
The $2$'s and $5$'s under the square roots seem very suggestive of the nature of $\operatorname{R}$.
Also, how could it be proved in that case?
Using
$$\frac{1}{\operatorname{R}(q)}-\operatorname{R}(q)=\frac{\left(q^{\frac{1}{5}};q^{\frac{1}{5}}\right)_{\infty}}{q^{\frac{1}{5}}(q^5;q^5)_{\infty}}+1$$
and
$$\frac{\eta (e^{-\pi\sqrt{n}})}{\eta \left(e^{-\frac{\pi}{\sqrt{n}}}\right)}=n^{-\frac{1}{4}},\, n\gt 0$$
(where $\eta (q)=q^{\frac{1}{12}}\prod_{n\ge 1}(1-q^{2n})$ is the Dedekind eta function),
I've been able to evaluate $\operatorname{R}(-e^{-\pi})$ and $\operatorname{R}(e^{-2\pi})$, but I don't know how could it be used to evaluate $\operatorname{R}(e^{-\pi})$. Perhaps something else is necessary.
I was inspired by Ramanujan's first letter to Hardy, where Ramanujan's $7\text{th}$ theorem states that
$$\cfrac{1}{1+\cfrac{e^{-\pi\sqrt{n}}}{1+\cfrac{e^{-2\pi\sqrt{n}}}{1+\cfrac{e^{-3\pi\sqrt{n}}}{1+\ddots}}}}$$
can be exactly found for any $n\in\mathbb{Q}^{+}$.
 A: Another approach is to use the identity $$\frac{1}{R(q)}-1-R(q)=\frac{f(-q^{1/5})}{q^{1/5}f(-q^5)}\tag {1}$$ where $$f(-q) =\prod_{n=1}^{\infty} (1-q^n)\tag{2}$$ Also note that the above is essentially Dedekind eta function $$\eta(q) =q^{1/24}\prod_{n=1}^{\infty}(1-q^n)\tag{3}$$ Thus in order to evaluate $R(q) $ for $q=e^{-\pi} $ we need the values of $\eta(e^{-\pi/5})$ and $\eta(e^{-5\pi})$. In fact what is needed is the ratio of their values and not the individual values.
We can note that if $k$ is the elliptic modulus corresponding to $q$ and $K$ is the corresponding complete elliptic integral of first kind then $$\eta(q) =2^{-1/6}\sqrt {\frac{2K}{\pi}}k^{1/12}k'^{1/3}\tag{4}$$ Also it should be noted that if $k$ is replaced by $k'$ then $K$ is replaced by $K'$ and $q$ is replaced by $q'$ where $(\log q) (\log q') =\pi^2$. Clearly the values $q=e^{-\pi/5},q'=e^{-5\pi}$ satisfy this equation and hence using a similar equation for $\eta(q') $ we get $$\frac{\eta(q)} {\eta(q')} =\sqrt{\frac{K} {K'}} (k'/k) ^{1/4}$$ and $\pi K'/K=-\log q=\pi/5$. Thus we get $$\frac{\eta(q)} {\eta(q')} =\sqrt{5}\cdot(k'/k)^{1/4}$$ where $k$ is the modulus corresponding to $q=e^{-\pi/5}$. It is well known from the value of Ramanujan class invariant $G_{25}=\phi$ (golden ratio) that $$2k'^2=1-\sqrt{1-\phi^{-24}}=1-12\sqrt{161\sqrt{5}-360}$$ and $$2k^2=1+\sqrt {1-\phi^{-24}}=1+12\sqrt{161\sqrt{5}-360}$$ and hence $$\frac{\eta(e^{-\pi/5})} {\eta(e^{-5\pi})} =\sqrt{5}\left(\frac{1-12\sqrt{161\sqrt{5}-360}}{1+12\sqrt{161\sqrt{5}-360}}\right)^{1/8}=A\text{(say)}\tag{5}$$ It should now be observed that if $q=e^{-\pi}$ then the value of right hand side of equation $(1)$ is $A$. Thus $R(e^{-\pi}) $ is the root of the equation $$\frac{1}{x}-1-x=A$$ or $$x^2+(A+1)x-1=0$$ so that $$R(e^{-\pi}) =x=\frac{-(A+1)+\sqrt{(A+1)^2+4}}{2}$$ where $A$ is given by equation $(5)$.
It requires some patience to get a simple expression for the above root though. Fortunately Ramanujan provided a denesting formula which helps us here. I reproduce it from one of my previous questions here: $$ \sqrt[8]{1\pm\sqrt{1-\phi^{-24}}} = \frac{-1+\sqrt{5}}{2}\cdot \frac{\sqrt[4]{5}\pm 1}{\sqrt{2}}\tag{6}$$ and thus $$A=\sqrt{5}\cdot\frac{\sqrt[4]{5}-1}{\sqrt[4]{5}+1}$$ It is now easy to verify the desired form for $x$ using a little symbolic algebra with the symbol $a$ given $a>0,a^4=5$.
Another curious fact is that if we change the value of $A$ to $A'=5/A$ then the corresponding $x$ gives the value of $R(e^{-4\pi})$. This is generalized by Ramanujan as a relation between the values of $R(e^{-2\alpha}) $ and $R(e^{-2\beta})$ where $\alpha, \beta$ are positive with $\alpha\beta=\pi^2$: $$\left(\frac{\sqrt {5}+1}{2}+R(e^{-2\alpha}) \right) \left(\frac{\sqrt{5}+1}{2}+R(e^{-2\beta})\right)=\frac{5+\sqrt{5}}{2}\tag{7}$$
A: $R(q)$ is expressible in terms of radicals for every $q=e^{2\pi i \tau}$, where $\tau$ is an imaginary quadratic irrational in the upper half plane. OP already observed it for $R(e^{-\pi\sqrt{n}})$. It is easy to obtain an algebraic equation satisfied by $R(q)$, but unwinding into radical form is more difficult.

The Roger-Ramanujan continued fraction has many equivalent forms:$$R(q) = \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\ddots}}}} = q^{1/5} \prod_{n=1}^\infty \frac{(1-q^{5n-1})(1-q^{5n-4})}{(1-q^{5n-2})(1-q^{5n-3})}.$$
Writing $q=e^{2\pi i \tau}$. From the latter expression, it can be shown that $R$ is modular of level $5$, therefore $R$ and the $j$-invariant has an algebraic relation of degree $[\bar{\Gamma}(1):\bar{\Gamma}(5)] = 60$ (bar means quotienting out center):
$$\tag{*}  R^5 (R^{10}+11 R^5-1)^5j+(R^{20}-228 R^{15}+494 R^{10}+228 R^5+1)^3 = 0.$$
You are concerned with $R(e^{-\pi})$. Since $j(i/2) = 287496$, $R$ is a root of degree $60$ equations over $\mathbb{Z}$. $(*)$ factors over $\mathbb{Q}$, the octic $$R^8+14R^7+22R^6+22R^5+30R^4-22 R^3+22 R^2-14R+1$$ appears as one of the factors. Numerical evaluation will tell you that $R(e^{-\pi})$ is indeed a root of this factor, completing the proof.
The other $59$ roots of $(*)$ are values of $R$ at $\Gamma(1)$-orbit of $i/2$ which are $\bar{\Gamma}(5)$-inequivalent.
