We demonstrate genuine multipartite quantum entanglement of four photons in their orbital angular momentum degrees of freedom, where a high-dimensional discrete Hilbert space is attached to each photon. This can encode more quantum information compared to the qubit case, but it is a long-standing problem to entangle more than two such photons. In our experiment we use pulsed spontaneous parametric down-conversion to produce the photon quadruplets, which allows us to detect about one four-photon event per second. By means of quantum state reconstruction and a suitable witness operator we find that the photon quadruplets form a genuine multipartite entangled symmetric Dicke state. This opens a new tool for addressing foundational questions in quantum mechanics, and for exploration of novel high-dimensional multiparty quantum information applications such as secret sharing.