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Entanglement is a powerful concept with an enormous potential for scientific and technological advances. A central focus in modern research is to extend the generation and control of entangled states from few to many qubits, and protect them against decoherence. Optical photons play a prominent role as these qubit carriers are naturally robust and easy to manipulate. However, the most successful technique to date for creating photonic entanglement is inherently probabilistic and therefore subject to severe scalability limitations. Here we avoid these by implementing a deterministic protocol with a single memory atom in a cavity. We interleave controlled single-photon emissions with tailored atomic qubit rotations to efficiently grow Greenberger-Horne-Zeilinger states of up to 14 photons and linear cluster states of up to 12 photons with a fidelity lower bounded by 76(6)% and 56(4)%, respectively. Thanks to a source-to-detection efficiency of 43.18(7)% per photon we measure these large states about once every minute, orders of magnitude faster than in any previous experiment. In the future, this rate could be increased even further, the scheme could be extended to two atoms in a cavity, or several sources could be quantum mechanically coupled, to generate higher-dimensional cluster states. Overcoming the limitations encountered by probabilistic schemes for photonic entanglement generation, our results may offer a way towards scalable measurement-based quantum computation and communication.