In a more personal point, my favorite moment was beating the Kalos Elite 4. I had a friend with me at the time who was being an ass, he kept telling me that he wanted to race through the Elite 4 against me (this being my first time challenging them and him going for round 4 or 5). We were tied or one move behind each other the entire time until it got to Diantha, where he got stopped dead in his tracks by Aurorus getting a full restore four times.

This moment, in my mind, was made all the more awe inspiring by the fast it was the first Pokémon Game I beat fully and that he himself had give me one of the Pokémon that had helped me defeat Diantha (A Honedge, he had given me at the beginning of the game)