  Kepler-289 is a three-planet system containing two sub-Neptunes and one cool
giant planet orbiting a young, Sun-like star. All three planets exhibit transit
timing variations (TTVs), with both adjacent planet pairs having orbital
periods close to the 2:1 orbital resonance. We observe two transits of
Kepler-289c with the Wide-field InfraRed Camera (WIRC) on the 200" Hale
Telescope at Palomar Observatory, using diffuser-assisted photometry to achieve
space-like photometric precision from the ground. These new transit
observations extend the original four-year Kepler TTV baseline by an additional
7.5 years. We re-reduce the archival Kepler data with an improved stellar
activity correction and carry out a joint fit with the Palomar data to
constrain the transit shapes and derive updated transit times. We then model
the TTVs to determine the masses of the three planets and constrain their
densities and bulk compositions. Our new analysis improves on previous mass and
density constraints by a factor of two or more for all three planets, with the
innermost planet showing the largest improvement. Our updated atmospheric mass
fractions for the inner two planets indicate that they likely have
hydrogen-rich envelopes, consistent with their location on the upper side of
the radius valley. We also constrain the heavy element composition of the outer
saturn-mass planet, Kepler-289c, for the first time, finding that it contains
30.5 $\pm$ 6.9 $M_{\oplus}$ of metals. We use dust evolution models to show
that Kepler-289c must have formed beyond 1~au, and likely beyond 3~au, and then
migrated inward.
