  Let $\boxplus$, $\boxtimes$ and $\uplus$ be the free additive, free
multiplicative, and boolean additive convolutions, respectively. For a
probability measure $\mu$ on $[0,\infty)$ with finite second moment, we find
the scaling limit of $(\mu^{\boxtimes N})^{\boxplus N}$ as $N$ goes to
infinity. The $\mathcal{R}$--transform of the limit distribution can be
represented by the Lambert's $W$ function. We also find similar limit theorem
by replacing the free additive convolution with the boolean convolution.
