What is the plot of following function

Question

What would the plot for $ x^{x^{x^{x^x...}}}$ look like for $x \in\mathbb{R}$?

I think all math-tools try to compute it recursively which end up with the error "standard computation time exceeded".

And sory, I could not express it well in the title.

Answer 1

It looks something like this:

where the even powers converge to the blue line and the odd powers converge to the yellow line. The point where these two converges to the same value is near $0.07$.

As pointed out in the comments, this function is not defined outside $[0,1]$ since the sequence $(x \uparrow\uparrow n)_{n=0}^\infty$ escapes to infinity.

Edit: This function also converges in $[1,e^{1/e}]$ according to StackTD's answer.

Answer 2

The infinite power tower $$x^{x^{x^{x^x{^{^{\cdot^{\cdot^{\cdot}}}}}}}}$$ converges only for $e^{-e} \le x \le e^{1/e}$, which is approximately in the interval $[0.0659 \,,\, 1.4446]$.

In that interval, the power tower can be rewritten in terms of the Lambert W-function which you can then use to have the graph plotted by computer software:

$$x^{x^{x^{x^x{^{^{\cdot^{\cdot^{\cdot}}}}}}}} = -\frac{W\left( -\ln x \right)}{\ln x}$$

You can also take a look at this plot by WolframAlpha.