  A dynamical system driven by non-Gaussian L\'evy noises of small intensity is
considered. The first exit time of solution orbits from a bounded neighborhood
of an attracting equilibrium state is estimated. For a class of non-Gaussian
L\'evy noises, it is shown that the mean exit time is asymptotically faster
than exponential (the well-known Gaussian Brownian noise case) but slower than
polynomial (the stable L\'evy noise case), in terms of the reciprocal of the
small noise intensity.
