  We consider different limit theorems for additive and multiplicative free
L\'evy processes. The main results are concerned with positive and unitary
multiplicative free L\'evy processes at small time, showing convergence to log
free stable laws for many examples. The additive case is much easier, and we
establish the convergence at small or large time to free stable laws. During
the investigation we found out that a log free stable law with index $1$
coincides with the Dykema-Haagerup distribution. We also consider limit
theorems for positive multiplicative Boolean L\'evy processes at small time,
obtaining log Boolean stable laws in the limit.
