I have a Nikon D5500 and Nikon SB800 speedlight. From previous answers it appears that it is not possible to do high-speed sync flash photography. Can I overcome the camera limitations?
The Nikon D5500 does not feature an Auto FP flash sync setting. This is the name Nikon uses for high speed sync. So although the SB800 does support AFP, your camera is not capable of using it.
The only practical way to overcome it would be to use a studio type flash that allows you to adjust the duration of the flash at constant output to longer than about 1/200 second or slower. These types of flashes are fairly expensive and probably cost several times the cost of upgrading to a Nikon body, such as the D7200, that does support AFP.
Not really. The D5500 is simply not capable of FP mode, so you can't do true high-speed sync, where the flash bursts are coordinated with the movement of the shutter curtains to illuminate the entire frame.
However, there is another technique, known as tail sync, where the flash is set off a little early, at full power. Power is actually a misnomer with speedlight output. The 'power' is determined by the length of the flash pulse. The longer the pulse, the higher the power. At full power, the pulse is actually long enough that it can "overlap" high shutter speeds. You generally have to be at full power on the speedlight, and at or above 1/1000s on the shutter speed to get this to work, and the window of opportunity is very narrow. This is even more of a power-suck than using HSS/FP flash, so it's not practical in a lot of applications.
To achieve tail-sync, you typically need a specific set of HSS-capable radio triggers. PocketWizard's TTL units call the feature Hypersync, Yongnuo YN-622s call it Supersync, and on the Phottix Odin, it's called Over Drive Sync (ODS). To really get the most out of it, the timing of the triggers needs to be adjustable, since different lights have different pulse patterns.
Tail-sync typically works better with studio lights, because they'll have longer light pulses, but it is possible with some speedlights at full power.