It may not be technically an 'Incredibly Tough', but it sure was a close call for me.
I had decided to see what the Fields of Valor augments might be like, so I attempted to search for something that would pose as a challenge (so I could get a potentially better augment) yet still could solo. Well, given anything above level 50 was out of the question due to their difficulty, I went with trying on my Scholar, which is still level 47 and was at the time.
I went looking at the FoVNM list on FFXIclopedia, and found that Balam-Agab looked to be an interesting shot. With that and given there was little information about it, I got a parchment and headed into Sauromugue Champaign.
Needless to say, even with full MP and HP, I was definitely not expecting the fight I got. I already had been fully prep'ed with a full Sublimation charge and as many buffs as I had possible as SCH/WHM (since it's a Couerl, figured I could easily dodge the Chaotic Eye by turning, which I did, but went with /WHM in case of Paralyze or other such). Well, after doing my usual debuffs on it that I could, I switched to Dark Arts. At the time, it was Lightningday. I pop'ed Parsimony and threw out a Stone II. The bugger absorbed it! ~120 HP healed. -.-; It also had Enwater on, as well. Well, needless to say, I couldn't use the Couerl weakness against it, so I had to resort to using Fire II and Blizzard II as my main damage points. It hit fast, and I could barely keep up. At the end of it, I had to use Tabula Rasa, switching to Light Arts to Penury+Celerity Cure III myself up to full before using Sublimation then going back to Dark Arts to pelt it with Parsimony+Alacrity Blizzard IIs. I eventually won, albeit with maybe 25-50% HP remaining.
Hardest battle I've ever fought while on Scholar, I must say. Quite exhilarating, too! Got a new lesson in the study of 'Things aren't always what they seem.'