Howard Shore, without question. Nobody has ever invested themselves in a project the way he did for "Lord of the Rings". The sheer breadth and depth he covered in creating distinctive themes for characters and cultures, blending them with a deft touch to create audio imagery I have never seen before. The Fellowship themselves had their own theme, separate and definite from everything else in the film. You had grand landscapes and intimate moments, echoes of ancient battles, strains of powerful love, poignancy in the death of kings. He even turned an inanimate golden ring into a character itself, full of temptation and evil. When you hear a theme repeated, it's with deliberate intent and an altered instrumentation for a definite purpose. John Williams is great and has earned his reputation, but my one criticism of him is that he tends to become repetitive between works such as in "Raiders of the Lost Ark" and "Star Wars". But that's a minor quibble and nothing more.
Howard Shore bows to no one.