Final answer:
The 'non-possessing class' refers to a social group below the elite, encompassing non-wealthy workers and non-Muslims who performed manual labor, lacking the genealogy or wealth that would afford higher status.
Step-by-step explanation:
The term "non-possessing class" as described in the Great Towns article refers to the social group that is positioned below the merchant ruling class. Members of this class include those who do not belong to the elite, such as townspeople, artisans, clerks, and other workers. They are referred to as 'non-possessing' because they do not own significant property or have wealth like the merchant class, and they cannot trace their genealogy to the Shirazi Muslims which would have given them higher status. Moreover, non-Muslims often comprised an even lower social tier within the towns, fulfilling roles such as servants or engaging in other types of manual labor.