With a very simple trick, it's possible to avoid unlocking the Energy Gauge on the game HUD. While the Energy Gauge appears to drain normally, there are other interesting side-effects that occur.
To perform the glitch, obtain 10 rupees and complete the game up until you are told by Sahasrahla to visit the Eastern Palace and have seen the cutscene showing the two switches that must be hit with the Bow.
Return to Ravio's Shop/Link's House. When Ravio asks what symbol you saw, give him an incorrect answer, allowing you to leave the house without the Bow. Now simply die and you'll warp back to Ravio, however this time he will ask if you want to rent back the bow for 10 rupees (even though you haven't rented it before). Accept the offer, and when you leave the house the game will not put the Energy Gauge on the HUD. The Energy Gauge will not be displayed on the HUD until you obtain the Stamina Scroll from the Ice Palace. However, it will overlap the Heart UI since the Heart UI is pushed to the right when you first get the Energy Gauge.
Aside from the nuisance of not having an Energy Gauge, there are a few other minor side effects. When you visit the Eastern Palace entrance, Osfala will not be there, allowing you to skip the encounter completely. If you visit Sahasrahla's house after completing the Eastern Palace, there will be two Sahasrahlas. The two Sahasrahlas say different things.