Is there a close form for the following nested radical?
$$\sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...}}}$$
It converges and
$$\quad \quad \lim_{n \to\infty} \sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...+\sqrt[n!]{n^n}}}}=1.8430759846682...$$
Is this number algebraic or transcendental?
Given the fact that neither Somos' quadratic recurrence constant, nor the nested radical constant are known to possess a closed form, I find it highly doubtful that this one will fare any better... Same as to the nature of the number, given that the nature of the afore-mentioned two constants is also unknown.