As luck would have it, a posting on Gizmodo lead me to a new product, the Digi-Slave Flex Ring 6400 LED Macrophotography Light. After some consultation with the manufacturer, I purchased one for testing.
To my great surprise and pleasure, this little beast pumps out enough light to permit videography not just at 300 fps, but also 600 and 1200 as well! Check out the video below, the shots were taken in a totally dark room, lit only by the Digi-Slave.
The trick to getting the maximum light out of this device is to use rechargeable Nickle-Metal-Hydride (NiMH) AA batteries. While these do not provide as much voltage as regular AA cells (1.2 vs. 1.5v), they have lower internal resistance and can provide higher current flows than even "high-discharge" Lithium batteries. Enjoy!