Adding a third celestial body to a gravitational system makes things wildly chaotic and unpredictable. Hence the 'Three Body Problem'. Some beautiful examples of periodic 3-body systems:

(via instagram user data__is__beautiful)

@infobeautiful I wonder if there are actually any such systems in the real universe.
@ondrej_zizka @infobeautiful I feel like the chance for any of them to (naturally) become stable like in the video over a long time would be extremely low.