We study a large class of potentials on the line with superquadratic growth at infinity and with the additional time-periodic dependence. We show that for a classical particle in such potentials most (in Lebesgue's sense) motions are quasiperiodic and in particular all motions are bounded, with no smallness assumptions on time-dependence. The class of potentials includes polynomial, exponential and much more. These results answer some questions posed by Littlewood in mid 1960s. Along the way we develop machinery for estimating the action-angle transformation directly in terms of the potential. We introduce also some apparently new identities involving singular integrals.