There are a number of optional side quests available which will help you progress in the game:
Rudy, the Ordon Village blacksmith who lives outside of town in a small cave, could use the hero's help in obtaining more water for his basin so he can continue his smithing. Doing so will yield a bottle, the ability to upgrade your sword at a later time, and Rudy's thanks.
Gaira, the groundskeeper who's often seen around Ordon Village, is distressed by the poisoned trees that have been reported around the Faron Woods. Upon investigation, she also discovers something she didn't expect, the handsome lumberjack Deacon.
The Great Fairy of Courage needs a large quantity of Mystic Jade in order to restore her forest to its previously un-poisoned state.
The Great Fairy of Power needs a large quantity of Goron Amber in order to restore her mountain to its previous state.
The Great Fairy of Wisdom needs a large quantity of Alchemy Stone in order to restore the frozen north to its previous state.
The Great Fairy of Kindness needs a large quantity of Goddess Plume in order to restore her lake to its previous state.
The Great Fairy of Magic needs a large quantity of Subrosian Ore in order to restore the underworld to its previous state.
The Great Fairy of Understanding needs a large quantity of Magic Crystal in order to restore the Great Fairy Palace to what it once was.
The Cave of Ordeals is a very deep trial that will test the hero's mettle and reward with some very special clothing. Find it at Death Mountain south of Septen Heights.
A shopkeeper in Kakariko had his prized bottle stolen and would like some help retrieving it from the thief, who he saw run into the graveyard.
The Island Palace in North Hyrule houses the Boss Rush challenge where you can (optionally) face all of your previous bosses again!
Parapa Palace and Midoro Palace in the North are optional mini-dungeons that house very special items.
Additionally, there are two optional fetch quests that can be unlocked by talking to characters in the Northern region of Hyrule:
The Books Fetch Quest will be unlocked after talking to Isan in the Ancient Library. The thirteen books that need to be found are as follows:
Anouki Book
Deku Book
Gerudo Book
Goron Book
Hylian Book
Kakariko Book
Kasuto Book
Ordon Book
Rauru Book
Rito Book
Tokay Book
Zola Book
Zora Book
The Plants Fetch Quest can be unlocked by speaking to a Saria citizen who mention rare plants and flowers and their potion making ability. The sixteen plants that would need to be found are:
Baba Bud
Calatian Corydalis
Hyrulian Deku Flower
Exotic Flower
Flame Flower
Frost Flower
Gnarled Flower
Goponga Flower
Hylian Hibiscus
Light Spirit Lavender
Lon Lon Milkweed
Mystical Flower
Pikit Plant
Sea Flower
Simple Flower
Town Flower
After having played through this game twice, and doing some thorough personal cartography the second time...I have not been able to locate the Zola Book or Zora Book anywhere. Do these really exist in the game??? I am skeptical... What is the initial trigger that allows the Lumberjack Love quest to be initiated by talking to Gaira? I was able to do this quest on my first playthough (though I didn't discover it until close to the end of the game), but I've beaten the game on my second file and she still doesn't say anything about poisoned trees. The Piece of Heart from this side quest is the only one I'm missing on my intended-to-100%-complete second file.
Apart from the above three objects (and whatever prize Isan might give me for completing the books quest), I believe I have completely acquired/finished everything in the game--I have every known item, all quests marked as complete (just no Lumberjack Love quest appearing at all), all 16 rare plants, and the previously implied 11 books and 31 Pieces of Heart. (It still only says 95% completion, though, which is lower than I expected.) Four other minor questions:
-In the Swamp Palace (where you get the Shovel) there's this left-hand room with two switches and a sunken treasure chest in the room below that. I activated both switches using the push-block and Hebra's Cane, and it made noises like something happened, except nothing did actually happen. The chest is still sunken, and nothing else appeared anywhere in the dungeon. Is there a way to get that sunken chest?
-Does the Kakariko City Amphitheater actually do anything? I expected it to be somehow involved in a side quest or the trading sequence, but that never turned out. What about that random guy Warun standing next to what looks like a flame geyser just to the left of the Amphitheater area?
-How is the Castle Town archery game actually supposed to work? It completely bewildered me, but it didn't seem like I was missing any significant prize by skipping out on it.
-Why are a few bookcases in the Hylian Library still unreadable, even though I've beaten the game?