Pulse-Width Modulation (PWM) is the method microcontrollers use to set LED brightness and the position of servos, so it is a pretty well-understood technique. This 16-Channel 12-bit Interface lets you set these PWM signals to 16 devices through the convenient 2-wire I2C interface. Set your pin assignments over an I2C call, and forget it - the chip will keep generating the signals until told otherwise.

This breakout board is configured to allow up to 62 live in a single chain, letting you run up to 992 PWM outputs. That's a lot of servos/LEDs under your control