I suspect that this thread exists mainly for the purpose of keeping the contentious "objective/subjective" controversy alive for those who can't get enough of repeating themselves. I'll try to avoid falling into that trap by offering a caution: the term "profound" in art is commonly used to mean nothing more than "very moving to me." If that's all it's going to come down to - and that would be consistent with the view that all artistic judgments are valid only for the individual making them - then we can all just agree that "profundity" in art is essentially a vacuous and useless notion, and save ourselves and each other some time we can spend more profitably elsewhere.
Personally, I don't happen to think that "profundity," as applied to art, is merely a pretentious way of referring to a personal emotional reaction. I think some works of art are capable of conveying much more complex meanings and resonances - both emotional and intellectual - than others, meanings and resonances which, although they won't be identical for all perceivers, can be seen and explored. For example, people will never stop exploring Shakespeare's plays and Wagner's operas - the books keep being written and published - while there will never be a need to say a word about vast numbers of works that give every bit as much pleasure. The profundity in art lies not in "how much it means to me" - i.e., how much I like it - but in the range of meanings, and in what kinds of meanings, it is seen to be capable of conveying and provoking.
This view does of course assume that art actually has the capacity to mean something, with the corollary that the range of meanings a given work will convey is determined to a major degree by the nature of the work - which is to say, that meaning is not merely something imposed by the audience. I must therefore call the following statement (from post #6 above) of no value to our understanding:
"I'm fine viewing profundity as a deeply personal experience. I can analyze works I feel are profound--Tristan und Isolde, Neon Genesis Evangelion, War & Peace, Paradise Lost, Vertigo, Mozart's 41st Symphony, etc.--until the cows come home, but at the end of the day such analysis doesn't matter much unless one feels it on a gut level."
Analyses of works of may have little immediate value to those who don't "feel it on a gut level," but one would be a fool to dismiss the perceptions of thoughtful individuals whose deeply felt observations might awaken us to things we hadn't realized were there until we were given a key with which to unlock their secrets. Anyone who's achieved any significant level of appreciation of any art has surely benefitted from the insights of others. Personally, I owe a tremendous debt of gratitude to thinkers who have assisted in my awakening to depths of meaning I might otherwise have come to perceive more slowly, if all.