One of the many secrets to the success and prevalence of insects is their versatile, robust, and complex exoskeleton morphology. A fundamental challenge in insect-inspired robotics has been the fabrication of robotic exoskeletons that can match the complexity of exoskeleton structural mechanics. Hybrid robots composed of rigid and soft elements have previously required access to expensive multi-material 3D printers, multi-step casting and machining processes, or limited material choice when using consumer-grade fabrication methods. Here we introduce a new design and fabrication process to rapidly construct flexible exoskeleton-inspired robots called flexoskeleton printing. We modify a consumer-grade fused deposition material (FDM) 3D printer to deposit filament directly onto a heated thermoplastic base layer which provides extremely strong bond strength between the deposited material and the inextensible, flexible base layer. This process significantly improves the fatigue resistance of printed components and enables a new class of insect-inspired robot morphologies. We demonstrate these capabilities through design and testing of a wide library of canonical flexoskeleton elements; ultimately leading to the integration of elements into a flexoskeleton walking legged robot.