I have been trying to find the exact value of $\log_2(e+\log_3(e+\log_4(e+...)))$. Using Desmos, I know it is approximately $1.95924$. I tried using WolframAlpha by using the Nest function but was unable to get it working properly because I am pretty new to WolframAlpha.
What I want to know is how I would solve it mathematically? I don't really know where to start so any help is appreciated.
If we know $\,a_n\,$ for certain $\,n \gg 1,$ then $a_{n-1}=y(a_n, n),\;$ wherein $$y(x)=\dfrac{\ln(e+x)}{\ln(n)} = \dfrac{1+\ln(1+x/e)}{\ln(n)} \approx \dfrac{{1+\dfrac xe-\dfrac{x^2}{2e^2}}}{{{\ln(n)}}}, \tag1 $$ Since denominators and numerators of $(1)$ changes slowly, then asymptotically $a_{n-1}\rightarrow a_n.$
Assuming $y= x$, easily to get the table of the approximated and exact estimations $$a_{n}\approx a_{n-1}\approx x\approx y\approx e\sqrt{e^2\ln^2 n - 2e \ln n+3} - e\ln n -1)\tag2$$ Exact solution has the closed form via analytic continuation of the Lambert W-function and has not essential differences from $(2).$
Iterations from the starting value $a_{20}$ (Table1) to $a_2$ (Table2) are shown at the right columns (at the inverse order).
Wolfram Alpha suffices all described calculations.