Many doubt Galadriel's abilities because she's female and just an "elf" I tend to get this a lot from some arguments...she's always the "healer, the wise one one, or just the "strategist" though that could be a male attribute as well (if we go with streotypes).
Some people say that even Glorfindel is more powerful than Galadriel, notably coz he killed a Balrog (though died with them), yet you have a quote that states, "a queen she was of the woodland Elves, the wife of Celeborn of Doriath, yet she herself was of the Noldor and remember the Day before days in Valinor, and she was the mightiest and fairest of all the Elves that remained in Middle-earth."
Now the complication starts with the term "greatest. When Tolkien says greatest and talks about a specific or a few set of skills, I think he really means that that particular individual is surpassed by none in that skill. For example, Tulkas someone mentioned long time ago that Tulkas was the greatest in strength and deeds in prowess (Valaquenta). So, as far as the Valar is concernd, Tulkas was the greatest strength-wise but that does not mean he was the wisest or the one with great inherent power. But when Tolkien uses the term "greatest" in the general sense, I think he means EXACTLY that they are the greatest - as a whole, not just talking about one thing here and there..just like when he referred to as the greatest maiar spirit being Sauron, and Feanor, greatest of the Noldor.
Hence, Sauron was the greatest of Morgoth's camp, then Feanor, greatest of the Noldor, which means just that, their skills as a whole, their inherent power and their abilities were not surpassed. No one surpassed Sauron as Morgoth's servant, and out of the Noldor, Feanor was best. Hence, when Tolkien saysSo, when it says that Galadriel is the greatest, save for Feanor (with a "Maybe"), that's what it means, I think!
I don't see how hard it is to understand that of the Noldor Galadriel was second or equal only to Feanor (maybe). Her attributes are not just one or two things there, but a match a match for everyone. She could match the loremasters of the Eldar (lore and wisdom, not to mention what she learned from a Maiar - Melian - who could do as she pleases in ME), and she was also a match for their athletes (so strength and body). She was great magically and power wise (the Mirror of Galadriel, and her dismantling the walls of Dol Guldur). Also was a student of the Maia Melian, where she learned many things from, and I guess, when called for, she was not afraid of taking armed combat if you take her defense of her mother's kin literally. Hence, she was not simply the greatest in one or two areas, but she was a great individual as a whole. And though we never know what she exactly did in defense of Lorien when it was assaulted three times, it did clearly say that "despite the valor of the Elven people, the power that dwelt there was too great to overcome unless Sauron came there himself (presumably with the One Ring on his finger). Saruman, on the other hand, had their powers restricted by the Valar, and perhaps were only as powerful as they were allowed to be (like when Gandalf fought with the Balrog). The Istaris are of Maiar origin, but they are NOT in their full Mair powers nor could they just use their magic as sparingly.