Consider directed polymers in a random environment on the complete graph of size N. This model can be formulated as a product of i.i.d. N×N random matrices and its large time asymptotics is captured by Lyapunov exponents and the Furstenberg measure. We detail this correspondence, derive the long-time limit of the model and obtain a co-variant distribution for the polymer path. Next, we observe that the model becomes exactly solvable when the disorder variables are located on edges of the complete graph and follow a totally asymmetric stable law of index α∈(0,1). Then, a certain notion of mean height of the polymer behaves like a random walk and we show that the height function is distributed around this mean according to an explicit law. Large N asymptotics can be taken in this setting, for instance, for the free energy of the system and for the invariant law of the polymer height with a shift. Moreover, we give some perturbative results for environments which are close to the totally asymmetric stable laws.