We report on the observation of a conspicuous loss in an ultracold mixture of ${}^7$Li and ${}^{88}$Sr atoms confined in a far-off-resonance optical dipole trap. We attribute the trap loss to the three-body inelastic Li-Sr-Sr collision and extract the corresponding three-body recombination coefficient $K_3$ at $T\sim 18.5,45,70,600\,\si{\micro K}$. The measured three-body recombination coefficient is about two to three orders of magnitude larger than the typical values convenient for realizing quantum degenerate gases, and essentially rules out the prospect of realizing ${}^7$Li and ${}^{88}$Sr mixtures of high phase space density. The measured three-body recombination rates agree with the unitarity limit at high temperatures. Our results also confirm that the three-body recombination loss is dominated by the light-heavy-heavy process, in agreement with the prediction for a system with large mass imbalanced and negligible intra-species interaction.