The anharmonicity of a transmon qubit, defined as the difference in energy level spacing, is a key design parameter. In transmons built from hybrid superconductor-semiconductor Josephson elements, the anharmonicity is tunable with gate voltages that control both the Josephson energy and the weak link transparency. In Sn-InAs nanowire transmons, we use two-tone microwave spectroscopy to extract anharmonicity ranging in absolute value from the transmon charging energy E_c to values smaller than E_c/10. This behavior contrasts with the predictions of the multi-channel short-junction model, which sets a lower limit on anharmonicity at E_c/4. Coherent operation of the qubit is still possible at the point of the lowest anharmonicity. These findings demonstrate the potential of quantum circuits that benefit from widely electrically tunable anharmonicity.