Consider the set of isoenergetically degenerate integrable Hamiltonians with two degrees of freedom. We show that a cusp-generic perturbation of a generic Hamiltonian in this set gives rise to meandering invariant tori - embedded Lagrangian tori which are not graphs. Moreover, an exponentially dense subset of perturbations admits higher order meandering tori, of all orders from two to infinity. These infinite order meanders have an endless nested structure.