I have found a number. Google and OEIS come up blank: 0.696340872970033948754981...

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.