Battle Chain is a running tally of defeated monsters. The tally resets to 0 upon exiting to the world or city map, upon leaving a dungeon by completing a quest, or upon being ambushed. A higher Battle Chain boosts art learning and upgrading (see Developing Arts), and generally improves the EXP multiplier for other stats. The maximum amount of chains that can be displayed by the game is 9999.
There is another term often confused with the Battle Chain: the Kill Count. This is the amount of defeated monsters in the current battle and is used to determine Monster Drops.