Battles aren't mini-games. They're tied into the main storyline. In other words, it's the main game of the whole DPP game, in which it won't be a mini-game. However, if you would still consider it a mini-game, I would select battling as my favorite. The nature of battling is very complex, and it takes wits to build that perfect team, outsmart your opponent, and win the game. There are a big variety of Pokemon and strategies to use, such as using the unexpected. It gets you hooked into a battle once you start, and you never want to stop until you prove yourself as the superior one.
Treasure digging would come in second. It's not as complicated as competitive battling is, but it is quite intriguing to interact with your friends in a new cave-like world. Also, the fact that the underground would give you free items if you find them is a definite plus, too. It also gets you hooked on because I sometimes can't find myself stopping the game for a minute until I find the item that I'm looking for.
The Safari Zone games and Pokemon Contests come in third. They're not quite as addicting as the other mini-games, but they can be pretty fun once you get the hang of it. I'm always trying to find some rare Pokemon like Skorupi in the Safari Zone, and I'm definitely working on the Dress-up part of the Pokemon Contests, since I've never been good at that division.
