Only a few additional parts will be needed. Resistors, capacitors. But do you realise that you will limit the LEDs to only 6 colours? Red, green, blue, yellow, cyan, magenta, white.There is a much easier way which is a far more simple circuit and will allow you to mix thousands of colours. Google for ws2812b LEDs or apa106 LEDs.
Well, first of all, it would help if you read the "How to use this Forum" at the top of every Forum.UPDATE: used to be at the top!Then you would know things like:We can't read your mind: what shift register? Part number please.XY Problem potential, here: Can you give us more info on what it is you wish to achieve?A schematic would be nice -- otherwise it's difficult to give specific advice.And, PaulRB's suggestion of "ws2812b LEDs or apa106 LEDs" is probably good, but, once again, not knowing the ultimate goal, hard to say.