We give an explicit analytic series expansion of the $(\max , +)$--Lyapunov exponent $ \gamma(p)$ of a sequence of independent and identically distributed random matrices in this algebra, generated via a Bernoulli scheme depending on a small parameter $p$. A key assumption is that one of the matrices has a unique eigenvector. This allows us to use a representation of this exponent as the mean value of a certain random variable, and then a discrete analogue of the so-called light-traffic perturbation formulas to derive the expansion. We show that it is analytic under a simple condition on $p$. This also provides a closed form expression for all derivatives of $\gamma (p)$ at $p=0$ and approximations of $ \gamma (p)$ of any order, together with an error estimate for finite order Taylor approximations. Several extensions of this are discussed, including expansions of multinomial schemes depending on small parameters $(p_1,\ldots,p_m)$ and expansions for exponents associated with iterates of a class of random operators which includes the class of so called topical operators. Several examples pertaining to computer and communication sciences are investigated: timed event graphs, resource sharing models and heaps models.