Single-photon detectors are ubiquitous devices in quantum-photonic-based communication, computation, metrology, and sensing. In these applications, $N$-fold coincidence photon counting is often needed, for example, to characterize entanglement. However, $N$-fold coincidence photon counting typically requires $N$ individual single-photon detectors and associated bias and readout electronics, and these resources could become prohibitive if $N$ goes large and the detectors need to work at cryogenic temperatures. Here, to break this limit on $N$, we propose a device architecture based on $N$ cascaded photosensitive superconducting nanowires and one wider nanowire that functions as a current reservoir. We show that by strategically designing the device, the network of these superconducting nanowires can work in a synergic manner as an $n$-photon detector, where $n$ can be from 1 to $N$, depending on the bias conditions. We therefore name the devices of this type superconducting nanowire multi-photon detectors (SNMPDs). In addition to its simple one-port bias and readout circuitry, the coincidences are counted internally in the detector, eliminating the need for external multi-channel, time-correlated pulse counters. We believe that the SNMPDs proposed in this work could open avenues towards conveniently measuring high-order temporal correlations of light and characterizing multi-photon entanglement.