Is there any other category of aircraft in the FAA regulations besides 'Ultralight' that does not require pilot licensure?
Yes, there is at least one. This probably was not what you had in mind, but the FAA has been very explicit about the following, for a variety of reasons.

Unmanned aircraft are explicitly regulated as aircraft. However, certain model aircraft require no certificate if operated under the provisions of §101. This is in contrast to all other unmanned aircraft which do require some sort of certificate or exemption, Small Unmanned Aircraft Systems regulated under §107 for example.

The following definition is from 14 CFR 1.1:


  Aircraft means a device that is used or intended to be used for flight in the air.


The following is excerpted from the FAA's FAQ page on Unmanned Aircraft Systems:


  What is an unmanned aircraft system (UAS)?
  An unmanned aircraft system is an unmanned aircraft and the equipment necessary for the safe and efficient operation of that aircraft. An unmanned aircraft is a component of a UAS. It is defined by statute as an aircraft that is operated without the possibility of direct human intervention from within or on the aircraft (Public Law 112-95, Section 331(8)).