We consider a particular generalized Lambert function, $y(x)$, defined by the implicit equation $y^β= 1 - e^{-xy}$, with $x>0$ and $ β> 1$. Solutions to this equation can be found in terms of a certain continued exponential. Asymptotic and structural properties of a non-trivial solution, $y_β(x)$, and its connection to the extinction probability of related branching processes are discussed. We demonstrate that this function constitutes a cumulative distribution function of a previously unknown non-negative absolutely continuous random variable.