  We report the discovery of two super-Earth mass planets orbiting the nearby
K0.5 dwarf HD 7924 which was previously known to host one small planet. The new
companions have masses of 7.9 and 6.4 M$_\oplus$, and orbital periods of 15.3
and 24.5 days. We perform a joint analysis of high-precision radial velocity
data from Keck/HIRES and the new Automated Planet Finder Telescope (APF) to
robustly detect three total planets in the system. We refine the ephemeris of
the previously known planet using five years of new Keck data and high-cadence
observations over the last 1.3 years with the APF. With this new ephemeris, we
show that a previous transit search for the inner-most planet would have
covered 70% of the predicted ingress or egress times. Photometric data
collected over the last eight years using the Automated Photometric Telescope
shows no evidence for transits of any of the planets, which would be detectable
if the planets transit and their compositions are hydrogen-dominated. We detect
a long-period signal that we interpret as the stellar magnetic activity cycle
since it is strongly correlated with the Ca II H and K activity index. We also
detect two additional short-period signals that we attribute to
rotationally-modulated starspots and a one month alias. The high-cadence APF
data help to distinguish between the true orbital periods and aliases caused by
the window function of the Keck data. The planets orbiting HD 7924 are a local
example of the compact, multi-planet systems that the Kepler Mission found in
great abundance.
