Electromagnetic showers developing from the collision of an ultra-intense laser pulse with a beam of high-energy electrons or photons are investigated under conditions relevant to future experiments on multi-petawatt laser facilities. A semi-analytical model is derived that predicts the shower multiplicity, i.e. the number of pairs produced per incident seed particle (electron or gamma photon). The model is benchmarked against particle-in-cell simulations and shown to be accurate over a wide range of seed particle energies (100 MeV - 40 GeV), laser relativistic field strengths ($10 < a_0 < 1000$), and quantum parameter $\chi_0$ (ranging from 1 to 40). It is shown that, for experiments expected in the next decade, only the first generations of pairs contribute to the shower while multiplicities larger than unity are predicted. Guidelines for forthcoming experiments are discussed considering laser facilities such as Apollon and ELI Beamlines. The difference between electron- and photon seeding and the influence of the laser pulse duration are investigated.