ArduinoUNO/MEGA, Is there a way to exceed 256 in analogWrite
Section 16, "16-bit Timer/Counter1 with PWM" of Atmel-8271J-AVR- ATmega-Datasheet_11/2015 describes using Timer/Counter-1 for 16-bit PWM, including sample code in both C and assembler. It might take a couple of reads over that section to get familiar with the hardware and learn what else T/C-1 is used for, and that you might have to give up while using it for PWM, but the code is fairly straightforward. 

Using this T/C would give you 2-bits more resolution in your PWM output than mapping onto an 8-bit counter, if that is what your project needs. 

+1 for the question. Not having needed PWM output for anything more than a demo with my DMM (yet), it hadn't occurred to me to look for a higher resolution until now.