There's a bunch of buttons at the top and eventually you'll level up and you'll be able to click this button. It's the Cloudwinder Dungeon and it's a 100 story tower-type dungeon with increasing difficulty with each level you obtain. The little red circle will tell you how many "free" attempts you have to progress.
As you climb upward, every 5 levels will reap greater rewards!
Here's a snip from an actual battle. It starts getting insane around level 35.
When the battle is over, reap your rewards! It has loads of gold and XP inside! Unfortunately, you only can collect each level once. However, you can reset it for free once from the main dungeon menu or use the Cloudwinder Reset Card that will appear in your backpack. (Not sure where it came from though.)
Elemental Tower Edit
With only half the floors, the Elemental Tower opens to level 60s to collect elemental particles and enhance skill in the elements. These are used to improve elemental attack and defence in your formation.