Reservoir Computing is a bio-inspired computing paradigm with the main advantage of being less computationally demanding than state-of-the-art neural networks. However, the performance of reservoir computing is limited comparing to other power-hungry Machine Learning algorithms. Deep Reservoir Computing addresses this problem by using multiple interconnected reservoirs to provide greater capacity for processing information, leading to improved performance. In this work we propose a photonic Deep Reservoir Computer: we test it on the speech recognition task, and we experimentally confirm its effectiveness. We show how some design choices can simplify the realization of a Reservoir Computer. Our photonic system is the first experimental Deep Reservoir Computer that can process speech recordings in real-time. We thus hope that this work can pave the way for efficient dedicated hardware for high-speed signal processing.