The introduction of parity-time (PT) symmetry in optics and photonics has initiated intense activities exploring the exotic properties of these structures, eventually leading to the more general notion of non-Hermitian photonics. Efforts to understand the behavior of these systems have revealed a host of distinct features originating from the unusual character of their eigenspectra and eigenstates. These include for example, spontaneous symmetry breaking, bandgap merging, laser self-termination, unidirectional invisibility, and ultra-sensitivity to external perturbations. A central notion pertinent to all these effects is the concept of exceptional points (EPs). Also known as branch points, EPs are non-Hermitian spectral singularities that arise when two (or more) eigenvalues and their corresponding eigenstates become identical. While exceptional points of order two have been studied thoroughly at both the theoretical and experimental level, higher order EPs are attracting attention only recently. Here we discuss a systematic approach based on a recursive bosonic quantization scheme for generating discrete photonic networks that exhibit exceptional points of any arbitrary order. We also discuss the spectral properties and the extreme dynamics near these singularities as well as their physical implementation in various photonic platforms.