It kind of looks like $\log(2)$, but it isn't.
This is a fixed point of a map $f:\Bbb{R}\to\Bbb{R}$ with $f(x) = (1-x)^{1-x}$. When you iterate the map it misbehaves at $0$ and $1$, so I started at $1/2$. Convergence is geometric, so it's not hard to find many decimals. I'm using Python $3$ with gmpy2.
My question is how do I proceed? I guess I'm putting it out here to see if someone can find some nice formula for this value. Tried posting to reddit but that got nuked real quick.
If this gets nuked for not being a question or unproductive or whatever, please advice how I should post!
Best Answer
Via a substitution this value is also $1-u$ in the following equations $$u^u=1-u$$ or $$u=\frac{\ln(1-u)}{\ln(u)}$$ At the end of the day it's just a number - among uncountably infinitely many others. It's not surprising at all that we don't have a way to express it in terms of a finite number of constants and functions we happen to have given names and notation.