We show how photonic quantum computation can be made more efficient by encoding several qubits within a single photon using its spatial structure. Instead of assigning one qubit to each photon, we use many spatial modes of the same photon to build large entangled cluster states while keeping the number of photons small. Using this approach, we generate cluster states equivalent to more than nine qubits using only two photons, with a detection rate of about 100 Hz. We also demonstrate that operations can be performed faster because qubits encoded in the same photon can be processed simultaneously. This work illustrates how high-dimensional encoding can reduce experimental resources and simplify measurement-based quantum computation.