Utilizing \sim 50 ks of Chandra X-ray Observatory imaging , we present an analysis of the intracluster medium ( ICM ) and cavity system in the galaxy cluster RBS 797 . In addition to the two previously known cavities in the cluster core , the new and deeper X-ray image has revealed additional structure associated with the active galactic nucleus ( AGN ) . The surface brightness decrements of the two cavities are unusually large , and are consistent with elongated cavities lying close to our line-of-sight . We estimate a total AGN outburst energy and mean jet power of \approx 3 \mbox { - - } 6 \times 10 ^ { 60 } erg and \approx 3 \mbox { - - } 6 \times 10 ^ { 45 } ~ { } { \mbox { $erg$ } ~ { } { \mbox { $s$% } ^ { -1 } } } , respectively , depending on the assumed geometrical configuration of the cavities . Thus , RBS 797 is apparently among the the most powerful AGN outbursts known in a cluster . The average mass accretion rate needed to power the AGN by accretion alone is \sim 1 ~ { } \mbox { $ { M _ { \odot } } ~ { } { \mbox { $yr$ } ^ { -1 } } $ } . We show that accretion of cold gas onto the AGN at this level is plausible , but that Bondi accretion of the hot atmosphere is probably not . The BCG harbors an unresolved , non-thermal nuclear X-ray source with a bolometric luminosity of \approx 2 \times 10 ^ { 44 } ~ { } { \mbox { $erg$ } ~ { } { \mbox { $s$ } ^ { -1 } } } . The nuclear emission is probably associated with a rapidly-accreting , radiatively inefficient accretion flow . We present tentative evidence that star formation in the BCG is being triggered by the radio jets and suggest that the cavities may be driving weak shocks ( M \sim 1.5 ) into the ICM , similar to the process in the galaxy cluster MS 0735.6+7421 .