A XP lamp (tier 1) is an item obtained during the Broken Home quest. It is the first lamp obtained in the quest, located in a chest found in the room with the book puzzle. It cannot be obtained on the first playthrough, and requires at least level 10 in each skill. Unlike other items, it can be taken out of the mansion. However, it cannot be banked. Rubbing it gives 1,000 XP in a skill of your choice.
Quest experience lamps