I know that the initial condition of the full double pendulum being horizontal yields motion that is not chaotic. Is there another set of initial conditions that would yield non-chaotic motion? Or do all non-horizontal initial conditions result in chaos?
Periodic orbits are possible for low energies (for which the outer limb does not flip) and very high energies (eg both limbs performing circular motion around the fixed point). In between there is a transition stage (quasi-periodic motion) towards chaos, followed by a transition back to periodic motion. I have not come across any statement of initial conditions defining the boundaries between each region, even for the 'simple' double pendulum (equal masses, equal lengths).
The following student project report gives examples of the transition from periodic to chaotic motion and back :