A way cool function, W(z), defined:

f(W) = We^{W} ⇒ W(f(z)) = z

As a sort of golden ratio of exponentials, the omega constant:

ω = W(1) = 0.56714...

Satisfies:

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:

x^{xx...} = -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)