For me, the first thing that forced me to whip out the ol' Zelda Dungeon cheat guide was the Ancient Cistern. In hindsight, it really wasn't as difficult as I thought it was, but I kept getting stuck in the underground area with the vined columns, spinning walking wheel and all the zombie bokoblins. I finally just needed some help! But the things that were difficult to get myself to do them were definitely the silent realms. I can't handle things chasing me, so that was like hell, pretty much!
I also agree that Demise was really tough. Whether you were close by him or far away, he always attacked... I always had to sacrifice some hearts just to drink a potion... it was tough! But I was so proud because I beat him my first time, no cheats! Granted, I used all of my potions and he had me down to one heart left... I was so excited when I finally got him, because I knew that if he hit me one more time, it was over! Such excitement!