The Big Easy is one of America's most unique cities. Drop most. What's wrong with saying one of America's unique cities?
Unique is, on its own, a potent word, and it must never be accompanied by an intensifier, since modifying it saps its considerable power. When you use unique, put it out there
alone—otherwise, say unusual.
Unique belongs to a group of words called absolutes or incomparables. Examples include dead, equal, essential, eternal, opposite, supreme. Such words resist being modified. Modifiers like more, most, absolutely, rather, and very either strip them of their strength or result in foolishness.
"Would you say 'very one-of-a-kind'?" asks Roy Blount Jr. in his book Alphabet Juice. Adding very or absolutely to unique, Blount says, "is like putting a propeller on a rabbit to make him hop better."