Quantum entanglement is important for emerging quantum technologies such as quantum computation and secure quantum networks. To boost these technologies, a race is currently ongoing to increase the number of particles in multiparticle entangled states, such as Greenberger–Horne–Zeilinger (GHZ) states. An alternative route is to increase the number of entangled quantum levels. Here, we overcome present experimental and technological challenges to create a three-particle GHZ state entangled in three levels for every particle. The resulting qutrit-entangled states are able to carry more information than entangled states of qubits. Our method, inspired by the computer algorithm Melvin, relies on a new multi-port that coherently manipulates several photons simultaneously in higher dimensions. The realization required us to develop a new high-brightness four-photon source entangled in orbital angular momentum. Our results allow qualitatively new refutations of local-realistic world views. We also expect that they will open up pathways for a further boost to quantum technologies.