Sure, fairly standard.
You need to multiplex the LEDs.
Say you can common cathode parts.
You drive the Anodes from a PWM pin each.
The cathodes are all pulled low using shift registers like TPIC6D595 (or 6B595 or 6C595, check avnet.com for availability).
You will analogWrite the PWM value LED1, and takes its cathode low for small time, then back high.
Repeat for as many RGB LEDs as you have.
The longer you have each on, the brighter the overall display will appear.
If each is on for 1/40 of a second, they will all appear to be on without flickering.
So, 3 outputs, 3 more to control the string of shift registers, and whatever buttons or analog inputs you want to control program flow on the remaining pins.