When you reach the open ended part of Final Fantasy XIII you can take on monster hunting missions. Daisuke Inoue, Final Fantasy XIII Battle Planner, designed these side quests with four rules.
1.) They should never force the player characters to equip any specific accessories.
2.) They should never require players to use TP abilities* to defeat them.
*(TP abilities are special skills such as Quake, Libra, and summons.)
3.) They should never require the player to use healing items to defeat them.
4.) They should never require the player to use shrouds* to defeat them.
*(Shrouds or smokes are buffing items you can use before encounters.)
“I can assure you that the mission monsters were designed first and foremost to present players with a real challenge,” said Inoue. “So much so, in fact, that our producer Mr. Kitase even came to us complaining he could not beat some of them and had no idea what to do, despite fighting with none of the handicaps mentioned above.”
Monster missions are the main after-the-credits-roll content in Final Fantasy XIII. Expect a challenge, if you decide to keep playing.