While playing with Desmos today, I typed the equation $y=x^{xy}$ and the graph came out to be 
I clicked the Learn More option given near my equation and Desmos said: Sometimes the calculator detects that an equation is too complicated to plot perfectly in a reasonable amount of time. When this happens, the equation is plotted at lower resolution.
What is the complication?
Best Answer
We have
$$\begin{align}&y=(x^x)^y\\ \implies &y^{\frac 1y}=x^x\\ \implies &y^{-1}\ln y=x\ln x\\ \implies &-y^{-1}\ln y^{-1}=x\ln x\\ \implies &y^{-1}\ln y^{-1}=-x\ln x\\ \implies &\ln y^{-1} e^{\ln {y^{-1}}}=-x\ln x\\ \implies &W\left(\ln y^{-1} e^{\ln {y^{-1}}}\right)=W\left(-x\ln x\right)\\ \implies &\ln y^{-1}=W\left(-x\ln x \right)\\ \implies &y^{-1}=e^{W\left(-x\ln x\right)}\\ \implies &y=e^{-W\left(-x\ln x\right)}\end{align}$$
This implies, your function is a non-elementary function and can be written as
$$f(x)=e^{-W\left(-x\ln x\right)}.$$
Then, note that $W(x)$ is real for only $x≥-\frac 1e$.
This means , we have
$$\begin{align}-x&\ln x≥-\frac 1e,\thinspace x>0\\ \implies &x\ln x≤\frac 1e \\ \implies &\ln x e^{\ln x}≤\frac 1e\\ \implies &W\left(\ln xe^{\ln x}\right)≤W\left(\frac 1e\right)\\ \implies &\ln x≤W\left(\frac 1e\right)\\ \implies &0<x≤e^{W\left(1/e\right)}≈1.3211\end{align}$$.