You would have to compromise too much, Three monitor for browsing and single for gaming is OK, but you are pushing it too far. SLI will just add fuel to the fire, since the GPU's not only need to send more data to the Screens but will have to sync between them too. The 7770 is good for 1650x1050, 1920x1080 at reduced settings, no memory hogs like AA or AF past 4x.
Going for three monitors you are looking at atleast 50% above Full HD(assuming 1366x768 screens, 3X for Full HD). So its still a bad idea.
So 3 monitors at 1366x768, A single 7770, this is what you can expect:
-Gaming at all 3 monitors:
Medium-High settings with Medium textures, no Post-Processing, no MSAA only FXAA, AF at 2X for current games.
About 2 years from now there will be games that wont even be playable at lowest settings with all monitors.
It will work at highest settings for most games from before 2011, assuming they support the Ultra Wide Wide resolution(not a typo).
-Using the three monitors during desktop mode, for multitasking, Gaming on only the middle one:
there will be no change from how you do things now. the gpu more than enough for desktop mode, and for games it does the same computation that it does on one monitor.
CrossFire(SLI is nvidia) 2 cards: All three monitors
Any game that gets less than 20 FPS on one card will have serious trouble with microstuttering, so it wont be worth anyway.On the rest of the games you will get around 50% improvement in raw FPS(down from the usual 70-80% that most dual card solutions get). You will still need to turn off AA, and AF to 4X, and you cant use ultra textures, just High or less.