Using Little's law to find the average N of customers I'm trying to understand Little's law and can't seem to get past this scenario:

Let's say people arrive at the register at 12 per hour. It takes 10 minutes (1/6 of an hour) to service each one of them. The law says that the long-term average number of customers is 12 * 1/6 = 2. After the first hour, my assumption is that there will be 6 customers not serviced. Then after the second hour, there will be 6 more. Why is the long-term average 2 and not the storeCapacity?

Is my usage of the law wrong because I'm not talking about a stable system?
Your intuition was right.  As you hinted, Little's law only applies to a stable system.  This is not a stable system.  The arrival rate exceeds the rate at which customers can be serviced.  Therefore, the number of customers waiting in line will grow without bound, getting increasingly larger as time increases.  That means this is not a stable system.

You can only apply Little's law to stable systems.  Since this isn't a stable system, Little's law doesn't apply.