There are four heart containers in Adventure of Link. To increase your life beyond this, you will need to level up.

1. Go south from Parapa Palace and into the single patch of grass surrounded by forest. Fight the Goriya and get the heart container.
2. After getting the hammer, break the boulder south of Rauru Town, then break the boulder that blocks the cave to the east of the path. Inside, there is a heart container.
3. When you get the Boots, go east from Mido Town and walk onto the ocean. After walking about two thirds of the way toward the palace, start trying to go north. Once you are able to go north, go to the right until you reach an action screen. It has a heart container.
4. East of Old Kasuto there are three rocks where you reveal the hidden palace. Walk along the southeast coast of the desert here to find a hidden action screen that has the last heart container.