Measurement Counts are a property of any feature within the inspection tab. They are used to set the desired number of measurements for a feature. When trapping to that feature you can measure until the desired count is reached and then the trapping will be automatically iterated to the next feature in the inspection list.
The default count for all types can be set to a value other than 0 using the Default Global Clout.
Each specific geometry type can also have a default set. This dialog will reflect the current default for geometry and will updated if the default is edited from a particular feature.