Thanks for the response Ryan, however in the video, the controller clears the aircraft before they call with any sort of emergency. It's immediately after contacting him he is cleared for a high speed climb.
The 7110.65 has no provision for approving an aircraft to exceed 250 knots below 10,000. As Ryan said, pilots and controllers are taking advantage of the ambiguity of the words "high speed climb" and "approved as requested." This is not a clearance to exceed speed limits. However, controllers aren't sky police and only see ground speed.
91.117 permits aircraft to exceed speed limits if the minimum safe speed for any particular operation is greater than the the speed limit. The 7110 has this to say about it:
14 CFR Section 91.117 permits speeds in excess of
250 knots (288 mph) when so required or recommended in
the airplane flight manual or required by normal military
operating procedures.
91.703 provides an exemption for civil aircraft outside the US (I believe that's 12nm offshore), but mentions you must still comply with certain sections of 91.117 if conditions exist.