It has been claimed that axions predicted in string theory may have a scalar potential with a much shallower region than the conventional cosine potential, especially when the dilute instanton gas approximation is violated. We first show that such axions generically undergo prominent resonance instabilities without being disturbed by the cosmic expansion after the onset of the oscillation, based on both analytic argument and lattice simulation. We further show that because of these instabilities, string axions in various mass ranges lead to a generation of gravitational waves (GWs) whose spectrum has peaks at various frequencies determined by the mass scales, dubbed GW forest. This suggests the possibility to explore the string axiverse through future multi-frequency GW observations.