What do you do when apple denies your request for refund on items you did not purchase in the App Store?

Then who did?

Purchases cannot happen magically. You or someone with access to your account or payment method had to have initiated the purchases. If you for example, downloaded a free app, it may have contained a free trial that would have renewed into a paid subscription unless you cancelled it before the trial ended.

Also, as mentioned, Apple reserves the right to deny any refunds. They can make exceptions, but they are not obligated to grant you a refund as established in the terms of service of the iTunes & App Stores you agree to when using them. Link-> Legal - Apple Media Services - Apple

The most you can do is contact Apple support and plead your case. Link-> Contact Apple Support

There's not really anything you can do. All sales are final and refunds are on an exception basis.

I suppose you could try contacting customer service and seeing if you can escalate the matter but, unless it's for a large amount of money, it's probably not worth the time.