My nomination: THE CENTAUR, by John Updike.
I first read it when I was fifteen (the age of one of the protagonists). I was crazy about classical mythology then, and I enjoyed it–but I honestly didn’t “get it.”
I read it again in my twenties in college (as a requirement of an American Literature course), and again I enjoyed it. But I still didn’t “get it.”
I read it again in my forties (again, the age of one of the protagonists). And this time . . . Holy crap. BOY, did I get it!
So–any others?
Oh, tons. When I was a kid, my mom would often tell me that there were books I just wasn’t ready for, and that if I tried them again when I was older, I might like them. She was right, and not just about childhood.
Of course, I can’t think of any good examples at the moment. 
But one small example is the entire oeuvre of Winnie the Pooh. When I was a kid, I thought they were dull, dull, dull and boring. As an adult, I adore them, and my little girl thinks they’re dull, dull, dull and boring!
One that I liked as a kid and now *dislike as an adult (if I may be so bold) is The Giving Tree. When I was a kid, it was a beautiful story of the sacrifice of Twoo Wuv. Now it’s a painful saga of a selfish monster who allows his enabler to literally cut her limbs off for his ungrateful use.
*ETA: Although I’m not sure which time I “got” it. Maybe there are two “gets” to get.
Catch-22 by Joseph Heller. I read it in junior high school in a marathon session and just…didn’t…get…it. I found it repetitive, convoluted, aimless, and maddeningly absurd.
Now I realize–while most of the above is true–it is one of the most brilliant satirical novels of the 20th Century. There are others, but this one always sticks in my mind. I think you need to have the repetitive, convoluted, aimless, and maddeningly absurd experience of dealing with a large bureaucracy populated by people who put their own petty wants above the overall purpose of the organization in order to appreciate it fully, or indeed at all.
Stranger
The Street of Crocodiles. In fact, I haven’t read it in years, I first tried at 13 or so. Maybe I’ll get it now.
Great Expectations*. When I read it in high school, I hated it, but when I reread it in my 30s, I realized it was a truly great novel.
The first book in the Sweet Valley High series. I read it when I was seven because I could and because it would annoy my nine year old sister (I can read it and you can’t!), but I didn’t get it. Man oh man, once I hit middle school though, it was like Francine Pascal (or one of the many people writing under that name, was creating masterpieces just for me.
Oh, thank god it’s not just me who finds that book horrifying. Though in my case, I also hated it as a child.
Shel, why did you betray us so? 
As to the OP, I can’t think of many examples, though I know that it happened a lot. The only one that springs to mind at the moment is a really obscure book that my dad had, called “Bloodsport”. It’s about a man and his son who go camping, and the son is supposedly kidnapped by a group of bikers (but he actually ran away to join them). It’s a weird and dark book, with lots of drugs and violence and group sex. I think I read it for the first time at about 10 or so? I thought that it was ok at the time, but I don’t think I really got it.
To their credit, my parents never really tried to control my reading, though they would occasionally warn me that I wouldn’t like something that was too old for me.
This, plus A Tale of Two Cities.
I read The Scarlet Letter in high school English and saw nothing interesting about it. Then I read it in a class this past semester and it was a mind-blowing experience.
When I first read The Lion, the Witch, and the Wardrobe, I was an eight year old Jewish kid who’d never read the bible and had only a minimal knowledge of Christianity. Then I reread the book (shortly before the movie came out), a bit after having read the bible for the first time. My roommate was hugely amused when I realized, about halfway through, that Aslan is in fact something of a Jesus character. (Seriously, I did not pick up on it at all when I read the book as a kid).
C.S. Lewis’ space trilogy (Out of the Silent Planet, Perelandra, and That Hideous Strength). I tried reading them as a young teenager in 1980, when I was first seriously getting into science fiction. Did not get them at all - in fact I gave up after Perelandra because they were, to me, incredibly dull and confusing. I remember thinking, “This is the same guy who wrote The Chronicles of Narnia?”
Tried them again in 2001-2002, when I was 35, all three books this time, and found them to be some of the best books I’ve ever read.
The difference is simple: the space trilogy was intended for adult readers, not young readers. For one thing, at 35 I was finally old enough to pick up on the subtext; for another, by 35 I had enough science fiction and fantasy books under my belt that I better understood the conventions of the genres and didn’t need Lewis to spell everything out for me like he did in the Narnia books.
Really? I read that trilogy when I was about 17 and really enjoyed it, but then I was in the process of studying religion and philosophy at A-level at the time which kind of helped.
The Three Musketeers. I thought it was about sword fights and ass kickings, not an allegory about the separation of church and state.
Okay, it’s both.
He didn’t. It was a cautionary tale. Don’t be the tree.
Or don’t be the selfish jerk. Either way, really.
As was Casablanca, and yet no one gets it.
Stranger
Well, as some around here are so fond of pointing out in the teen sex threads, there’s a huuuuuuuuuuuuge difference between 13 (my age at the time) and 17 
Looking back at the list of books I’ve read (I’ve kept a record since 1979), I see that Out of the Silent Planet was literally the first science fiction book I ever read. From there, the next few books on my list go like this:
Harriet the Spy by Louise Fitzhugh
Perelandra
Lucky Starr and the Big Sun of Mercury ← My very first Asimov, whose stories I proceeded to devour over the next several years
A Spell for Chameleon by Piers Anthony ← My very first fantasy novel outside of Narnia
Hmmm … no, wait, looking at my list it appears I didn’t read The Chronicles of Narnia until 1982. That may not be accurate, though, because I know I read them through at least twice, so if the first time was before November 1979, I wouldn’t have a record of it. <looks at bookcase> As it happens I still have the original Narnia boxed set that I read way back then, and opening one of the books I see that they are from a 1977 printing. So I probably did read them before 1979.
So the space trilogy was my first foray outside of the standard “young reader” genre, where the primary characters were adults, not children/teenagers, and I probably just wasn’t ready for it yet. Looking over the list of science fiction, fantasy, and other books I read between 1980 and 1984, I probably could have handled the trilogy by age 17. But in early 1980 when I first tried, The Empire Strikes Back was still a couple months away, leaving the original Star Wars and a few episodes of Star Trek and Lost in Space as my only exposure to science fiction.
My parents gave me the Space Trilogy in 4th grade (“Look, more books by the Narnia guy!”), but I didn’t understand them or like them at all. In 12th grade, Out of the Silent Planet was on the reading list for English class, so I read it. I actually got it, and I’ve since read the other two. They’re much deeper in psychology and philosophy than the Narnia books, which is to be expected given the intended readers.
One of my college English teachers said he didn’t get While I Lay Dying until he was 30, but I still haven’t read that.
Hey, don’t feel bad. Even though I went to a Missouri Synod Lutheran elementary (weekly chapel on Wednesdays, prayer every day, religion the number one topic of study, although spelling, grammar, and math were also hugely important) and my third-grade teacher read the Magician’s Nephew and The Lion, The Witch, and The Wardrobe to us, I didn’t get the Christian imagery for ages after- maybe in sixth grade, when it seemed horribly obvious.
I read a fair amount of Roger Zelazny books when I was quite young ( The Amber series, Creatures of Light And Darkness, Lord of Light, others ) that while I enjoyed, I didn’t get much of what was going on. Because he liked really complicated plots; because he often didn’t spell out or resolve everything; because they took me so long to read I forgot the start before I finished; and because I just didn’t have the knowledge to get a lot of the references. It’s just not the same if, say, you don’t know anything about Hinduism or Buddhism or Egyptian mythology.
Crime and Punishment - Fyodor Dostoevsky - I read it first at university when I was terribly cool and sophisticated, and read a story about a guy who killed an old woman and could real off Russian character names and be really impressive. I read it again in my thirties and read an extraordinary novel in which a woman was killed, but it was so much more than a murder. Again in my 40s, and I read a book about emotions I could now identify with - and it really scared me. That is one hell of a book!
I read some Jane Austens when I was in High School and couldn’t stand them: ugh what a boring existence, yecch how mercenary, how pathetic, sheesh don’t BE like that get a life!
I had little sympathy for people who live in the world as it is rather than forcing the world to wrap around their notion of how things should be.
They got better.