When you have:
$$x=y^{y^{y^{y\dots}}}$$
You have:
$$x=e^{-W(-\ln(y))}$$
ONLY when the power tower converges.
But what about when it doesn't? Is there any way to justify $2^{2^{2^{\dots}}}=e^{-W(-\ln(2))}\approx 0.82467854614-1.56743212384$?
Can we evaluate it in some analytical way or something when it doesn't converge, sort of like divergent summations and principal values?
Also, if you are simply going to argue that $2^{2^{2^{\dots}}}=\infty$, then I argue that I am asking about a way to make the evaluation non-infinite, much like how Ramanujan's summation method allows us to evaluate $1+2+3+\dots=-\frac1{12}$.
While it doesn't make "sense" because we see that it diverges to infinity, I was wondering if we could apply a similar method so as to get a finite value when the infinite power tower diverges to infinity.
I don't think that there can ever be a justification of a finite value for the notation $2^{2^{2^\cdots}}$ . But in fact in most cases, when we talk about tetration we see the things different, namely in form of an exponential tower, which begins with some value $x_0$ and puts an exponential base to it. So that, for the case of tetration, we should always write: $$ \begin{array} {r} \large x_0\\ \large {\,_2x_0 }\\ \large{\,_{\,_2}} \large {\,_2x_0 }\\ {\,_{\,_{\,_\ldots}}} \large{\,_{\,_2}} \large {\,_2x_0 }\\ \vdots \end{array}$$ (but which is difficult to model in $ \LaTex $)
With this we can write our beloved infinite towers and setting $x_0$ to the fixpoint near $\small 0.82467854 + 1.56743212 î $ $\,^{[1]}$ has then even a sensical interpretation (and reflects the infinite , let's call it now for distinction from the powertower, "exponential-tower").
$^{[1]}$ which is what also WolframAlpha gives for $e^{-W(-\ln(2))}$