Recommendation for project needing ~15 potentiometers

Do you mean arduino 'd read 15 pot's and send it to PC via USB?
What is the voltage coming from pot's? If it's in 0- 5V range, than you need an analog multiplexer, build your own board with 2 x IC (8 inputs) or even one 16 inputs.
SFE has : Search Results for analog multiplexer - SparkFun Electronics Look around, now you know what to google...