While monitoring coms I recently heard a Cessna Centurion that had just cleared the runway request a "high speed taxi", which was approved by the tower. I'm assuming that means he wants to taxi at a faster than normal rate to his FBO of choice. Are high speed taxis defined somewhere as something you can request of ground control?
There's no definition in the AIM or regulations, as far as I know, but there is one in the P/CG, sort of (emphasis mine):
HIGH SPEED TAXIWAY− A long radius taxiway designed and provided with lighting or marking to define the path of aircraft, traveling at high speed (up to 60 knots), from the runway center to a point on the center of a taxiway. Also referred to as long radius exit or turn-off taxiway. The high speed taxiway is designed to expedite aircraft turning off the runway after landing, thus reducing runway occupancy time
So at least as far as the FAA is concerned, a high-speed taxi may be up to 60kts.
As for requesting it, you can request more or less anything. There's an active EAA chapter at my local airport and pilots who are in the process of building and testing aircraft sometimes request a high-speed taxi on the runway. The controllers don't question it or set any speed restrictions, it's all on the pilot. I suppose if things were busy they might deny the request, but the pilots know to do these things at quiet times anyway.
He was likely asking for approval to use the runway for a simulated take off without actually departing. Usually testing engine performance after modifications.
The FAA does not provide a taxi speed limit but most airlines require observance of SOP "standard operating procedures" as part of their certification process. Normal SOP's are 30kts max with 10kts preferred. The rule I have seen for most flight training schools is "no more than a fast walk".
While not specific to the FAA's FAR or AIM, this FAA video describes the design and requirements of high speed taxies and refers technical specifications to AC 150-5300-13A-chg1. They define a high speed taxi as having no more than 30deg and nor more turn radius than 1500ft.
The exact dimension depends on aircraft category, IFR minimums, whether a reverse turn is needed, etc. No speed requirements are required, they define the accommodation for speed as "higher than normal taxi speed".