  We present all scalar-tensor Lagrangians that are cubic in second derivatives
of a scalar field, and that are degenerate, hence avoiding Ostrogradsky
instabilities. Thanks to the existence of constraints, they propagate no more
than three degrees of freedom, despite having higher order equations of motion.
We also determine the viable combinations of previously identified quadratic
degenerate Lagrangians and the newly established cubic ones. Finally, we study
whether the new theories are connected to known scalar-tensor theories such as
Horndeski and beyond Horndeski, through conformal or disformal transformations.
