Update:
Before moving to a new home each Sim is allowed to leave no more than one object. 15 homes = 30 different objects.
These can be: a unique object purchased from Buy Mode, a unique career or party reward, a crafted object unique to a skill (i.e. one painting, one cooked dish, one gourmet dish, etc.) or an object unique to a collection (i.e., one fish, one metal, etc.).