iPhone 13 Pro Max doesn't have the option to update to iOS 16

Please check: Settings - General - iPhone Storage - Search here for the iOS update file. If found, delete the downloaded file, restart the phone and check again for the update. After restarting, You may see the option for iOS 15.7, or iOS 16.

Have you previously used a beta version of iOS? If so, remove the beta profile, restart the phone and check again for the update.

Also: charge the battery and preferably leave it connected to the charge circuit during the update process.

Also: turn off the Low Power Mode option.

Did you turn off Low Power Mode?

Have you used any beta version?

You are welcome, and thank you for the feedback.