So, you think I messed up in the title? I'm going to argue that the language police are wrong here, and common usage is perfectly fine.
The common language maven's take on this is that "unique" means "one and only; single; sole," and so simply cannot be qualified by "most" or "extremely."
Nonsense, I say, because, unqualified, "unique" applies to every single thing in the universe -- which is why it is a "single thing in the universe." Let's take one of the least differentiated classes of objects in the world, say, electrons. Still, every single electron is unique in some way -- it's a part of this atom and not that atom, or it's in this orbital and not that one, or, even given two electrons in the same orbital of the same atom, they have different spins. In other words, I could walk down the street and point at every single thing I see and truthfully say, "That's unique! That's unique! That's unique! That's unique!" It would be true, but pointless.
Therefore, when we say, "He's a unique guy," we are already saying, in essence, "He's an especially unique guy." In other words, while every human being is unique, this fellow stands out in that regard. So what is the problem adding on "very" or "extremely" to emphasize something's uniqueness even further? Nothing, I say! Modify away!