Sampling the distribution of bosons that have undergone a random unitary evolution is strongly believed to be a computationally hard problem. Key to outperforming classical simulations of this task is to increase both the number of input photons and the size of the network. We propose driven boson sampling, in which photons are input within the network itself, as a means to approach this goal. We show that the mean number of photons entering a boson sampling experiment can exceed one photon per input mode, while maintaining the required complexity, potentially leading to less stringent requirements on the input states for such experiments. When using heralded single-photon sources based on parametric down-conversion, this approach offers an ∼e-fold enhancement in the input state generation rate over scattershot boson sampling, reaching the scaling limit for such sources. This approach also offers a dramatic increase in the signal-to-noise ratio with respect to higher-order photon generation from such probabilistic sources, which removes the need for photon number resolution during the heralding process as the size of the system increases.