A XP lamp (tier 9) is an item obtained during the Broken Home quest. It is the ninth lamp obtained in the quest, located in a chest found on the 1st floor[UK]2nd floor[US], in the bathroom of the Family Chamber (Ingram's room). It cannot be obtained on the first playthrough, and requires at least level 90 in each skill. Unlike other items, it can be taken out of the mansion. However, it cannot be banked. Rubbing it gives 96,000 experience in a skill of your choice.
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