Does the NEC allow for smaller than 15 amp / 14 gauge circuits?
It looks like the NEC allows only specific circuit ratings. From the 2014 NEC:


  210.3 Rating
  Branch circuits recognized by this article shall be rated in accordance with the maximum permitted ampere rating or setting of the overcurrent device. The rating for other than individual branch circuits shall be 15, 20, 30, 40, and 50 amperes. Where conductors of higher ampacity are used for any reason, the ampere rating or setting of the specified overcurrent device shall determine the circuit rating.


The exception specified is if the circuit supplies only a specific item, an "individual branch circuit":


  Branch Circuit, Individual. A branch circuit that supplies only one utilization equipment.


There are also some other exceptions that don't apply here, like for motor loads and in industrial settings.