Non-classical states of light are of fundamental importance for emerging quantum technologies. All optics experiments producing multi-qubit entangled states have until now relied on outcome post-selection, a procedure where only the measurement results corresponding to the desired state are considered. This method severely limits the usefulness of the resulting entangled states. Here, we show the direct production of polarization-entangled photon triplets by cascading two entangled downconversion processes. Detecting the triplets with high efficiency superconducting nanowire single-photon detectors allows us to fully characterize them through quantum state tomography. We use our three-photon entangled state to demonstrate the ability to herald Bell states, a task which was not possible with previous three-photon states, and test local realism by violating the Mermin and Svetlichny inequalities. These results represent a significant breakthrough for entangled multi-photon state production by eliminating the constraints of outcome post-selection, providing a novel resource for optical quantum information processing.