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In this paper, we investigate the dynamic behavior of a modified Leslie-Gower predator-prey model with the Allee effect on both prey and predator. It is shown that the model has at most two positive equilibria, where one is always a hyperbolic saddle and the other is a weak focus with multiplicity of at least three by concrete example. In addition, we analyze the bifurcations of the system, including saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. The results show that the model has a cusp of codimension three and undergoes a Bogdanov-Takens bifurcation of codimension two. The system undergoes a degenerate Hopf bifurcation and has two limit cycles (the inner one is stable and the outer one is unstable). These enrich the dynamics of the modified Leslie-Gower predator-prey model with the double Allee effects.