High-quality lithium niobate (LN) thin-film microresonators provide an ideal platform for on-chip nonlinear optical applications. The strict phase matching condition should be satisfied for an efficient nonlinear optical process, which requires dispersion engineering with a LN microresonator. However, this is challenging in single microresonator, resulting from the fabrication error. Here, we demonstrate strong nonlinear effects in a photonic molecule (PM) structure composed of two strongly coupled lithium niobate microdisks. The fabrication flow of the device includes three steps: (1) femtosecond laser ablation in water to pattern a primary structure of the PM on the lithium niobate on insulator (LNOI) wafer, (2) focused ion beam (FIB) milling to define the boundary of the coupled microdisks, and (3) chemical wet etching to obtain free-standing PM. The size mismatch of the microdisks enables phase matching by employing coupling-induced frequency splitting to compensate for the material and geometric dispersion. With a continuous wave excitation, rich nonlinear optical phenomena including cascaded four-wave mixing and stimulated Raman scattering were observed around the second harmonic signal. Meanwhile, an ultra-high four-wave mixing conversion efficiency with a rough estimation of absolute conversion efficiency with 14% as obtained when the second harmonic signal power is at microwatts level. The coupled LN microdisk PM is of great potential for broad applications in nonlinear integrated photonics.