King's Favors are activities that randomly pop up during gameplay where the King requests something of the Adventurers. Favors can include a variety of things, from Local Items, to even living creatures, such as Wabbits. Completing Favors give you Local Item Value based on the task and the week number. Some are 10,000G times the week number, others are 15,000G times the week number, and so on. Only one favor will be active at a time, and one unlocks on each Chapter.
After killing the 2 big Monsters in Chapter 1, the King will ask you for his first favor; he want you to bring him either a Khan Combo or Holy Curry. The Local Item Value for accomplishing this task is 10,000G times the week number.
The first Sunday after Chapter 2 starts, the King will ask you for a Wabbit. The Wabbit can be found and fought (extremely rare) in the Spring Cave. This Wabbit will only drop the "Wabbit" item, and it auto-counters. Return the Wabbit item to the King to receive 10,000G times the week number in Local Item Value.
(Not known when it is requested), the King will ask for a Salamander Bug. The Salamander Bug can be found in Lava Cave on the bottom floor. There is an empty space with a small, dead tree in behind it. If you land on that space you will get the Salamander Bug. Take the Salamander to the King and he will reward you with (reward details unknown).