The nested radical $$1.75793\approx\sqrt{1+\sqrt{2+\sqrt{3+\cdots}}}$$ has yet to be given a closed form.
However, nested radicals of the form, $$\sqrt{A+B\sqrt{A+B\sqrt{A+\cdots}}}$$ have the closed form, $$\frac{1}{2}\left(B+\sqrt{B^2+4A}\right).$$ I decided to attempt putting the first nested radical into this form. I know that there are values of $A$ and $B$ that make this possible. For example, we could have $A\approx{0.604235}$, $B=\sqrt{2}$.
We rewrite $$\sqrt{1+\sqrt{2+\sqrt{3+\cdots}}}$$ as $$\sqrt{a}\sqrt{\frac{1}{a}+\frac{\sqrt{b}}{a}\sqrt{\frac{2}{b}+\frac{\sqrt{c}}{b}\sqrt{\frac{3}{c}+\frac{\sqrt{d}}{c}\sqrt{\frac{4}{d}+\cdots}}}}$$
Is it possible to find solutions for $A$ and $B$ such that $$A=\frac{1}{a}=\frac{2}{b}=\frac{3}{c}=\frac{4}{d}=\cdots$$ and $$B=\frac{\sqrt{b}}{a}=\frac{\sqrt{c}}{b}=\frac{\sqrt{d}}{c}=\frac{\sqrt{e}}{d}=\cdots$$?
Obviously this system of equations is false and therefore there are no solutions, right? (It would work if instead of using $\{1,2,3,4,\dots\}$ we used, for example, $\{1,2,8,128,\dots\}$). But why are there still values of $A$ and $B$ that work. Try plugging $A\approx{2.829018637}$, $B=2$ into $$\sqrt{\frac{1}{A}}\sqrt{A+B\sqrt{A+B\sqrt{A+\cdots}}}.$$ You get the nested radical constant $\approx{1.75793}$, which is what we want. So how can the system of equations be false and there still be solutions?