Multi-photon entangled states of light are key to advancing quantum communication, computation, and metrology. Current methods for building such states are based on stitching together photons from probabilistic sources. The probability of $N$ such sources firing simultaneously decreases exponentially with $N$, imposing severe limitations on the practically achievable number of coincident photons. We tackle this challenge with a quantum interference buffer (QIB), which combines three functionalities: firstly, it stores polarization qubits, enabling the use of polarization-entangled states as resource; secondly, it implements entangled-source multiplexing, greatly enhancing the resource-state generation rates; thirdly, it implements time-multiplexed, on-demand linear optical networks for interfering subsequent states. Using the QIB, we multiplex 21 Bell-state sources and demonstrate a nine-fold enhancement in the generation rate of four-photon GHZ states. The enhancement scales exponentially with the photon number; larger states benefit more strongly. Multiplexed photon entanglement and interference will find diverse applications in quantum photonics, allowing for practical realisations of multi-photon protocols.