From my understanding, a few Belgian beers use a second yeast strain during bottle conditioning. This is to impart a different and unique flavor to the final beer. In that case, that would be 'double fermentation'. I've never noticed a 'triple fermented' beer.
But I imagine that you have two choices when 'triple fermenting'.
One is to add a fresh batch of the same strain of yeast in at secondary. The advantage of this is that a freshly made batch of yeast would be very active and could drop the FG a few more points than the original yeast could. This would yield a dryer and more alcoholic final beer.
The second option is to add another strain of yeast at secondary. This would of course change the flavor profile. And potential affect the final gravity.
If the second option is taken and it is the same yeast as bottle conditioning, then you get no real benefit in my mind.
If the second option is taken and a third strain is added at bottle conditioning, then you might get something unique.
If the first option is taken you could at least get more alcohol, if that is your aim.
All in all, it sounds like a lot of work and cost for no real benefit.
The flip side to this argument is that I've had a lot of really good Belgian beers, so they probably know what they are doing. =)