  We present an analysis of deep Chandra X-ray observations of the galaxy
cluster MS 0735.6+7421, which hosts the most energetic radio AGN known. Our
analysis has revealed two cavities in its hot atmosphere with diameters of
200-240 kpc. The total cavity enthalpy, mean age, and mean jet power are
$9\times 10^{61}$ erg, $1.6\times 10^{8}$ yr, and $1.7\times 10^{46}$ erg/s,
respectively. The cavities are surrounded by nearly continuous temperature and
surface brightness discontinuities associated with an elliptical shock front of
Mach number 1.26 (1.17-1.30) and age of $1.1\times 10^{8}$ yr. The shock has
injected at least $4\times 10^{61}$ erg into the hot atmosphere at a rate of
$1.1\times 10^{46}$ erg/s. A second pair of cavities and possibly a second
shock front are located along the radio jets, indicating that the AGN power has
declined by a factor of 30 over the past 100 Myr. The multiphase atmosphere
surrounding the central galaxy is cooling at a rate of 36 Msun/yr, but does not
fuel star formation at an appreciable rate. In addition to heating, entrainment
in the radio jet may be depleting the nucleus of fuel and preventing gas from
condensing out of the intracluster medium. Finally, we examine the mean time
intervals between AGN outbursts in systems with multiple generations of X-ray
cavities. We find that, like MS0735, their AGN rejuvenate on a timescale that
is approximately 1/3 of their mean central cooling timescales, indicating that
jet heating is outpacing cooling in these systems.
