fun constants 1: s

$s=$1.776775040097054697479730744038756748637… is the unique
nontrivial real root of $s+1 = s^s$, and it’s got some fun properties. It is sort of like an exponential analogue of the golden mean, but not in the same way that the omega constant is.

$(s+1)^{s+1} = (s^{s})^{s^{s}} = s^{s^{s+1}} = s^{s^{s^{s}}}$

$(s+1)s = s^2 + s$
And also:
$(s+1)s = s\cdot s^{s} = s^{s+1} = s^{s^{s}}$
so:
$s^2 + s = s^{s^{s}}$

The import of this is that we can find interesting additive decompositions of tetrational towers of ${}^{n}s$ on some occasions. I think:
${}^n(s+1) = {}^{2n}s$