A potentiometer requires analog input, so if you want ~ 15 pots, your only choice for an Arduino without additional hardware is the Arduino Mega 2560. Note, the Mega only has 16 analog inputs, so it doesn't have room for growth.
Now, I was curious and did a google search for USB potentiometer, and found this. It is a little expensive, but it probably would allow to to use up to 16 or 24 pots without having to build a project with the Arduino: http://www.controlanything.com/Relay/Relay/UPOTS