A family of local diffeomorphisms of R(n) can undergo a period doubling (flip) bifurcation as an eigenvalue of a fixed point passes through -1. This bifurcation is either supercritical or subcritical, depending on the sign of a coefficient determined by higher-order terms. If this coefficient is zero, the resulting bifurcation is "degenerate." The period doubling bifurcation with a single higher-order degeneracy is treated, as well as the more general degenerate period doubling bifurcation where a fixed point has -1 eigenvalue and any number of higher-order degeneracies. The main procedure is a Lyapunov-Schmidt reduction: period-2 orbits are shown to be in one-to-one correspondence with roots of the reduced "bifurcation function," which has Z2 symmetry. Illustrative examples of the occurrence of the singly degenerate period doubling in the context of periodically forced planar oscillators are also presented.