B737 NG Captain reported rejecting the takeoff at 100 kts when #1 engine EGT exceeded the red line

Narrative

First flight of the day. First time crew flying together. All SOP were complied with on preflight planning. Thorough briefings were conducted to include the Rejected Take Off (RTO) briefing with the mechanical actions and duties of the RTO.Take off data and the final weight manifest was thoroughly reviewed for a hot day in ZZZ. Weight; temperature and altimeter were verified accurate on takeoff data. Cleared for takeoff on runway XXL full length. Thrust was set and the takeoff roll began. At 85 knots the number 1 engine EGT was 925 and continuing to rapidly increase. At 95 knots the number one engine EGT was 945. The EGT continued through the limitation of 950 up to 966 and the EGT gauge turned red. The rejected takeoff procedure was initiated simultaneously at the 100-knot call. Rejected takeoff procedures were accomplished. Crash fire and rescue evaluated the aircraft. The aircraft was taxied back to the gate. No injuries to passengers or crew. No aircraft damage. The aircraft was taken out of service. Great CRM amongst the crew.