For me, Fellowship of the Rings gets better with time. Your mileage will vary with this, and it will likely change depending upon what you bring to the book and where you are in your life; but it's that sort of book, too. It obviously does not change, yet I have as I've interacted with it over the years. Growing up, I loved the opening sections of the book in Hobbiton of the Shire, where life was simple and peaceful and folk lived in holes in the ground on the west side of the Brandywine; the old forest loomed over Buckland; and the big folk kept to themselves over in Breeland, but for the most part had no issues with their Hobbit neighbors. As I've grown up, I've loved the middle section of the book: the quest upon which Frodo and his three friends embark, eventually to meet Strider ("all that is gold does not glitter..."), a wandering and elusive ranger from the north; they carry with them a dark secret and are pursued by dangerous enemies into the wilds.
But now I find myself most enjoying the latter part of the book, after the party comes to Rivendell, when the fellowship proper is formed, and they embark upon their great quest. I enjoy the delve into Moria, and the escape across the Bridge; the stay in Lothlorien where time is different even if the days feel the same; and finally the slow journey down Anduin the Great, toward the falls, and toward the moment when the fellowship faces a difficult choice. It's such a sad ending to book, and the beginning of the next book is no less sad.
Some critique the book for a lack of "depth," but I think that is in part because of what they bring to the book. They bring with them ideas of what a story, narrative, and characters should be like. But this book is of another time and place; perhaps even a bit antiquated when it first released, it feels much older than it is, and this is especially true once the party leaves Rivendell. This is a book not about adventure, although there is adventure in it, but a book about lamenting things that have been lost or shall be. Every character throughout talks of things they or their people have lost because of the growing shadow in the east: the elves their connection to the world, men their great kingdom, the dwarves their mighty halls, and the halflings their peaceable lives. Because the magic of the world is fading, no one can win, or at least it seems as much; yet the characters push on because they must do what's right. This heroic impulse in the narrative is also perhaps off-putting for modern readers because we see how foolish it is. In a world where everything seems to be waning, the only thing waiting for Frodo is death; yet he goes forward anyway.