There are some possibilities why she wasn't offered an exchange:

If the items have been clearanced out, they will show up in our system as 'Not On File.' There is no way to return or exchange these items without a valid receipt because Target does not sell them anymore.

Or it could be that this woman was disrespectful towards an employee, and they refused to help her out by exchanging without a receipt. Some people don't realize how much their attitude can affect how far the employee is willing to help you out.

As for all the commenters, let this be a learning experience for everyone. If you are buying a gift, include a gift receipt, not just at Target, but wherever.