"The Judgment of Paris" is definitely not the only myth where a golden apple stirs up some trouble. Another famous one from Greek mythology is the incident where Heracles steals the golden apples of the Hesperides as one of his Twelve Labors. Then, of course, there's the story where the fast-running Atalanta loses a race because Hippomenes keeps throwing irresistible golden apples on the ground behind him and she can't help but pick them up. Golden apples also pop up in Norse mythology where they're kept by the goddess Idunn and provide the gods with their immortal youth. When sneaky Loki helps steal them, however, all of Asgard is in for a rocky time.
Grimms' Fairy Tales includes a story called "The Golden Bird" in which a... well... golden bird steals a golden apple from a king's golden apple tree. When the king's son sets off to nab the thieving bird he's in for a lot more than he bargained for. Click here for the whole story. There are lots of other European folk tales that kick off with a golden apple being stolen, including ones from Russia called "Tsarevitch Ivan, the Fire Bird and the Gray Wolf" and one from Serbia called "The Nine Peahens and the Golden Apples."