User blog comment:Edwin Shade/An Equation I Found/@comment-1605058-20171106160419

Interestingly, if you swap \(\pi^{1/e}\) and \(e^{1/\pi}\), you get a number which is a solution of \(x=(e^{1/\pi})^{(\pi^{1/e})^x}\). One of the solutions of this equation is exactly \(e\), but the power tower converges to the other solution, \(x\approx 2.382\).

Similarly, your number is the "wrong" solution to \(x=(\pi^{1/e})^{(e^{1/\pi})^x}\), whose other solution is \(\pi\).