Totally depends on the game.
The first game whose League I defeated was Gold, and I've been ruined ever since. It totally spoiled me by giving me a complete other region and 8 more badges to collect (in hindsight, GSC Kanto leaves a lot to be desired, but at the time it was mind boggling).
For other generations, I would strive to beat the Battle facilities; this was most fun in Emerald and Platinum/HGSS (they both had Frontiers rather than just Towers), and the PWT in BW2.
Come gen 6, I gained an interest in completing a Living Dex, and since then that's usually been my goal. Complete/update my living dex, and once that's done, get the game as prepped as possible for competitive. Build up money, TMs, items, so I can create and modify Pokemon for competitive as easily as possible.
I very rarely just put the game down once I've finished the story; but if I'm not liking the competitive scene (or where there IS no competitive scene like BDSP or Legends Arceus), I tend to leave it down after I've completed the Dex.