It was a beautiful golden harp, and when Thorin struck it the music began all at once, so sudden and sweet that Bilbo forgot everything else, and was swept away into dark lands under strange moons, far over The Water and very far from his hobbit-hole under The Hill.
J.R.R. Tolkien
There are a few works, just a few, which have given me glimpses of Fantasy.
Sure, I’ve read and seen and played plenty of fantasies. Few have shown me Fantasy. You see, Fantasy is a realm beyond our own: a mysterious, beautiful, dangerous place we are seldom privileged to see. Tolkien called it Faerie.
The realm of fairy-story is wide and deep and high and filled with many things: all manner of beasts and birds are found there; shoreless seas and stars uncounted; beauty that is an enchantment, and an ever-present peril; both joy and sorrow as sharp as swords. In that realm a man may, perhaps, count himself fortunate to have wandered, but its very richness and strangeness tie the tongue of a traveller who would report them. And while he is there it is dangerous for him to ask too many questions, lest the gates should be shut and the keys be lost.
There are worlds we know, the worlds of Tolkien and C.S. Lewis and The Legend of Zelda and Final Fantasy: Middle Earth, Narnia, Hyrule, Spira, Ivalice and others. None of these are Fantasy, yet all of them have given me glimpses of it. Like Thorin’s golden harp, they carried me to faraway places full of danger and beauty and mystery: snowy peaks and tangled forests and mines whose gems shine like stars in the dark heavens.
I enjoy escaping to Fantasy. My brief trips there are never planned, sadly. They just happen, and I think they’re a good thing. Consider these words from Tolkien:
I do not accept the tone of scorn or pity with which ‘Escape’ is now so often used. Why should a man be scorned if, finding himself in prison, he tries to get out and go home? Or if he cannot do so, he thinks and talks about other topics than jailers and prison-walls?
In my ordinary life, I appreciate my fleeting visits to Fantasy. It’s nice to get away!
Typewriter Monkey Task Force 
You have hit upon a very important point that I don’t think many people appreciate: it takes more than just having swords and dragons and magic to make a real, mythopoiec Fantasy. It takes at least glimpses of striking wonder, of a realm beyond beyond, a sense of touching something deeper and more powerful and wilder. C. S. Lewis said you could have dozens of stories set in space and they will never be Science Fiction, just the same love or crime or war stories that might as well be set on Earth. When Fantasy descends into genre, when adventure and magic become mere spectacle, it misses its highest nature, which is to point beyond and out.
You make an excellent point! For a story to be a proper fantasy — one that invokes Fantasy — its fantastical elements can’t be incidental, like a cheap coat of paint. In proper fantasies, the fantastical elements mean something, I think. A dragon in a true fantasy is more than a large reptile. In a proper fantasy, a dragon is wise or monstrous or evil, not just scaly and cold-blooded.