Using resonant inelastic x-ray scattering, we observe in the bilayer iridate Sr3Ir2O7, a spin-orbit coupling driven magnetic insulator with a small charge gap, a magnon gap of ≈92 meV for both acoustic and optical branches. This exceptionally large magnon gap exceeds the total magnon bandwidth of ≈70 meV and implies a marked departure from the Heisenberg model, in stark contrast to the case of the single-layer iridate Sr2IrO4. Analyzing the origin of these observations, we find that the giant magnon gap results from bond-directional pseudodipolar interactions that are strongly enhanced near the metal-insulator transition boundary. This suggests that novel magnetism, such as that inspired by the Kitaev model built on the pseudodipolar interactions, may emerge in small charge-gap iridates.