I'm not 100% sure about HD5xxx but on 6xxx you can use up to two non-DP outputs at the same time.
This is due to framing constraints. Chip has only 2 framing electronics blocks, which actually freame the picture ( send individual pixels at the correct rate and sequence, generate sync signals etc etc).
Native DisplayPort doesn't need framing as it operates more like network card. Picture is send down the cable as a series of packets and it it monitors job to assemble them in picture, unpack audio etc etc.
So, for third monitor you either need something with DP input or ACTIVE adapter DisplayPort to something else ( VGA or DVI usually).
"ACTIVE" part is important since you can get simple ones ( that are nothing more than two connectors and a piece of wire), which are passive and when used, GPU will internally turn its DP output into compatibility DVI/HDMI mode and use one of the internal framing cirucits for framing.
You need ACTIVE adapter, which takes DP and frames it. I believe Sapphire has one or two cheap solutions they offer as an optional accessories.If you plan to use more than 1920x1200@60Hz at that output, take care to use dual instead of single link DVI adapter.
I have three monitor setup on my Sapphire HD6850 It's a 3x Syncmaster 20" 1600x1200 setup. Two monitors are connected through DVI and for the third I have DP to VGA adapter and it is connected through analog VGA input.
I must say this was not the best of choice, but I had to use it, since it was only available at the time. VGA signal is not 100% sharp and all details have small horizontal shadow. Not immediatelly visible, but annoying at long hours of precision work and close range.
It is not even discernible for other users ( games, movies etc), but I would still go for active DP-DVI. Once again, avoid passive ones.
On a journey of life I chose the psycho path...