The one I remember most vividly was catching Kyogre after it had Rested in Emerald with a PokeBall. Also in Emerald, I remember sitting down fighting against Wallace and adjusting the volume of my GBA SP with the Champion music. There's this particular part where the melody places heavy emphasis on the first note and kind of dies away about 3 times, I think... I'd turn the volume to max at that emphasised note and then bring it down as the note died away... I remember thinking that I should totally record it and put in on YouTube. So glad I never got round to it xD
I also remember being so happy finally beating Glacia with my Blaziken 'Chicky', then getting whooped by Drake's Salamence... that was in Sapphire.
A more recent memory was catching a shiny Luvdisc in Black by chain fishing. It was pretty great (: