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The Golden Apple is an apple in Super Paper Mario that only appears once in the game. In The Overthere, after going through Door E, Mario, Luigi and Bowser will find Princess Peach fast asleep after she ate the apple. The apple is a reference to the forbidden fruit of the Garden of Eden, and is even called that by Luvbi. It may also be a reference to the immortality-granting golden apples that are said to grow at the end of the world in Greek mythology.
When they find Peach, Luvbi explains that the Golden Apple is so delectable that anyone who eats it will end up falling into deep sleep for 100 years. She also says that there is a taboo fruit around that may restore her consciousness. The fruit that Mario and his team must get is a Black Apple, the only fruit that wakes Peach. (Other fruits do wake her, but only temporarily and with humorous side effects, such as making her giant, giving her a mustache or changing her into an actual peach.)
It is impossible to eat this apple during regular gameplay, as the only one ever shown onscreen is tossed over the edge of the clouds by Luvbi after the heroes discover the sleeping Peach.