Office 365 - Unable to send/receive messages due to mailbox quota

When an Office 365 account/mailbox (NetID or service account) reaches its mailbox quota, it will no longer be able to send or receive messages. The mailbox quota calculation includes all email and calendar data (including any attachments) within the account/mailbox.

Mailbox quota cannot be increased.

There are three kinds of notifications when a mailbox is nearing, or at, capacity:

The user receives an email warning that the mailbox is approaching the maximum size limit. This warning is intended to encourage users to delete unwanted mail. Received when mailbox reaches 49 GB.
Prohibit Send
The user receives a prohibit-send notification email when the mailbox size limit is reached. The user cannot send new messages until enough email is deleted to bring the mailbox below the size limit. Received when mailbox reaches 49.5 GB.
Prohibit Send/Receive
Prohibit Send/Receive Exchange Online rejects any incoming mail when the mailbox size limit is reached, and sends a non-delivery report (NDR) to the sender. The sender has the option to try resending the mail later. To receive messages again, the user must delete email until the mailbox is below the size limit. Received when mailbox reaches 50 GB.

What can I do to keep my mailbox below my quota?

Important: There are no system backups for Office 365 accounts. Once a message is deleted and expunged from your recover deleted items location, it cannot be restored.

Note to IT support staff - one possible temporary solution is to create a service account to move data to which the customer does not access daily but cannot be deleted due to data retention policies/needs.