Nonlinear electronic transport has emerged as a powerful probe of the quantum geometry in topological quantum materials, where the band topology and broken symmetries facilitate power law current voltage responses beyond Ohms law. While nonlinear transport of the second and third orders has been studied in several quantum materials, higher-order transport has so far mainly remained experimentally inaccessible, leaving more detailed features of the quantum geometry unexplored. Here, we observe higher order nonlinear electronic transport up to the seventh harmonic order in multilayer magnetic topological insulator MnBi2Te4. We find an even-odd behavior where the odd order nonlinear transport components dominate while the even-order ones are suppressed. Temperature and magnetic field dependent measurements show a strong correlation between the nonlinear transport and the magnetic phases of MnBi2Te4. Through scaling analysis and theoretical calculations, quantum metric multipoles and nonlinear Drude conductivities are identified as the microscopic origins of the nonlinear transport.