To support extremely high data rates in 6G wireless networks, reconfigurable intelligent surface (RIS)-assisted terahertz (THz) communications have gained much attention in recent years. By manipulating the phase shifts of reflecting elements, the RIS can proactively adjust the wireless propagation environment of THz systems, thereby enhancing the overall throughput significantly. To realize the full potential of RIS-assisted THz systems, an acquisition of accurate channel information is of great importance. However, since the wavefront of the THz electromagnetic signal is spherical, the conventional channel estimation techniques using the planar wavefront assumption suffer from severe performance degradation in the near-field RIS-assisted THz systems. An aim of this work is to propose an efficient channel estimation technique for near-field RIS-assisted wideband THz systems. Key idea of the proposed polar-domain frequency-dependent RIS-assisted channel estimation (PF-RCE) scheme is to estimate the sparse multipath components (i.e., angles, distances, and path gains) of the near-field THz channel by exploiting the polar-domain sparsity and common support properties. We demonstrate from the numerical evaluations that PF-RCE achieves a significant performance gain over the conventional THz channel estimation schemes in terms of the normalized mean square error (NMSE).