We extend the effective field theory of inflation to a general Lagrangian constructed from Arnowitt-Deser-Misner variables that encompasses the most general interactions with up to second derivatives of the scalar field whose background breaks temporal diffeomorphism invariance. Degeneracy conditions, corresponding to 8 distinct types-only one of which corresponds to known degenerate higher-order scalar-tensor models-provide necessary conditions for eliminating the Ostrogradsky ghost in a covariant theory at the level of the quadratic action in unitary gauge. Novel implications of the degenerate higher-order system for the Cauchy problem are illustrated with the phase space portrait of an explicit inflationary example: Not all field configurations lead to physical solutions for the metric even for positive potentials; solutions are unique for a given configuration only up to a branch choice; solutions on one branch can apparently end at nonsingular points of the metric and their continuation on alternate branches lead to nonsingular bouncing solutions; unitary gauge perturbations can go unstable even when degenerate terms in the Lagrangian are infinitesimal. The attractor solution leads to an inflationary scenario where slow-roll parameters vary and running of the tilt can be large even with no explicit features in the potential far from the end of inflation, requiring the optimized slow-roll approach for predicting observables.