A way cool function, W(z), defined:
f(W) = WeW ⇒ W(f(z)) = z
As a sort of golden ratio of exponentials, the omega constant:
ω = W(1) = 0.56714...
e-ω = ω
Which is equivilent to a lot of fun things otherwise unsolvable in exponentials.
The Lambert W Function can be used to express the analytic result of the infinite power tower:
xxx... = -W(-ln x)/ln x
Its derivative is:
W(x)/[x(1 + W(x))]
And its integral cannot be expressed in terms of itself and elementary functions.
(From Eric Weisstein's MathWorld, http://mathworld.wolfram.com/LambertsW-Function.html)