CE 560 Single Pilot reported multiple avionics and communication failures after the PFD; radios and other avionics failed during climb. Eventually communication was re-established with ATC to obtain help and special handling for landing.

Narrative

Following takeoff from ZZZ and while climbing on the ZZZZZ departure the pilots PFD alerted to failure of the heading gyro. Reversion mode was selected and the departure route was continued. Approximately two minutes later Radio Management (RMU)1 and 2 failed with both displays going blank. This results in complete communications failure. Additionally; the emergency com radio display is now black. After approximately five minutes RMU #2 begins displaying a visible transponder code window. 7700 is entered and a reply indication is noted. After a minute I changed the code to 7600 to alert ATC to a com failure. Onboard the aircraft there is no ability to transmit or receive. Additionally; both audio panels have failed. The FMS and the Garmin 500 gps both have the flight plan route loaded and are functioning normally. An attempt is made to utilize onboard Wifi to contact ATC through an FBO email address but the wifi connection is also down. The onboard satellite phone is displaying a 'NO Service' screen. Climb is continued to the last assigned altitude of FL230. During the course of this climb consultation of the FMS fuel management screen reveals that without climbing to a higher altitude (Filed altitude is FL400)there will be insufficient fuel on board to fly the route and land at the clearance destination. The choices are to divert to another airport and ultimately descend and leave the cleared route or continue a climb to the filed altitude of FL400. The aircraft is flying through the congested corridor; now passing over the ZZZ1 area. Determining it unwise to descend and deviate to an airport in this high density area I determined that the best course of action would be to continue climbing to the filed altitude and maintain the flight planned and cleared route. Approximately 45mins into the flight RMU #2 began to display a flickering com frequency window. Due to the failure of both audio panels radio reception could only be heard over the cockpit overhead speaker. A transmission on 121.5 elicited a response from ZZZ Center who then assigned an appropriate communication frequency. Contacting them I explained the cascading avionics failure and advised that additional loss of com was a possibility. The flight continued on with a couple amendments to the route; deviations for weather near the destination and constant need to request ATC to repeat transmissions due to the difficulty hearing clearly over the cockpit speaker.Upon landing further investigation revealed that the emergency com (#3) was still inoperative when the battery switch was placed in Emer mode. Throughout the flight RMU #1 remained completely black; both audio panels remained inoperative and the primary PFD heading gyro remained failed necessitating continued operation in reversion mode. In summary; my decision to climb beyond my cleared altitude allowed the flight to continue on the route assigned and further allowed the flight to exit the northeast corridor at a relatively low density flight level without both a descent and route deviation in congested and sensitive airspace. I would note that at the time of this writing the manager of this aircraft has informed me of the discovery of a tripped circuit breaker in the 'J Box' accessed through the aircraft battery compartment in the tail baggage compartment of the aircraft. Resetting this breaker appears to have restored com function in the aircraft.