Depends on the game...
ALttP - I dealt the final blow to Agahnim in Hyrule Castle. I think I've beaten the game, and then... he draws me into the Dark World! Suddenly I realize I have 6 more dungeons. Kind of a shock the first time you play.
OoT - Meeting Sheik in the Temple of Time. He surprised me. I didn't know if he was going to attack me or what. But he turned out to be an ally, and taught me the Prelude of Light (and gave a hint about where to find the Hookshot, which I never would have found otherwise). Well, actually he's a she... oh, well.
MM - The Kafei sidequest. I had wondered what that purple-haired girl in a Keaton Mask who runs off suspiciously (actually a he, my gender errors never cease) was doing at the mailbox every morning on the first day ever since I started the game. You feel so happy for them, and yet so sorry for them in the end. And not just them, that poor autistic postman who won't leave because the delivery is still scheduled.
WW - Entering Hyrule Castle for the first time. At first the game felt like Link's Awakening, with the whole island thing. It didn't feel like I was in Hyrule... until I entered that castle. I heard the music, I saw the statue, pulled the Master Sword... and it was almost like coming home, in a weird way. It was just magical.
TP - Interestingly, my best memory of this game never actually happened except in a dream. I dreamed I was playing Twilight Princess, and I was in the Temple of Time. I know that Midna was gone, for some reason. When I came to the room where the Darknut mini-boss was, it looked different. It looked like the room with the Master Sword in OoT. I went to grab the sword, but as I approached it... a blue magical barrier appeared around it (similar to the red magical one you see when fighting Twilight creatures). I heard Midna's voice giggling, and something jumped out of my shadow. Link turned around quickly, and Dark Link was right behind him! So I fought him, but he was much stronger than in the Water Temple. He could use all of my items and moves. It was particularly annoying when he used a Back Slice, and he killed me once with a Finish blow (but I had a fairy). The fight lasted for hours (since he ALSO had a fairy and red potion), but eventually I beat him and took the sword. After I took it, the blue light appeared around Link like in OoT, and he appeared in the place where you normally get the Master Sword in the forest.