Aside from ARAM

Are their any league mini-games you like to play?

My friends and I enjoy a hardy helping of "25 kills first" in a bot games. It's fun KSing each other haha.

There's "Hide-and-Seek" using the Dominion map and "Catch The Teemo" using Summoner's Rift.

Draft racing is amusing, although it quickly gets one-sided with certain comps.

Basically, after draft picking, you're supposed to dive the enemy fountain to score points. You're not allowed to go near the minions for gold/exp.