Question

What games have higher levels when battling the Pokemon League for the second time?

Answer 1

It depends on when you battle them for the second time.
Levels never increase: RBY, GSC, RSE, DP, XY
Levels only increase after a certain in-game event: FRLG, Pt, HGSS
Levels increase upon second challenge: BWBW2, ORAS, SM, USM
source

Comments

The events in question being the Stark Mountain post-game, Sevii post-game, and defeating Blue?

---

I think so.