Using Outlook 2013 (32 bit) on Windows 10 Pro (all updates applied).
When Outlook 2013 is disconnected from Exchange (because of no internet) or Offline (by clicking the Offline button), the first email sent goes to the Outbox. Whilst still offline, several other emails are sent and sit in the outbox. However, when Outlook reconnects, the first email stays in the outbox, and all other email send as normal. This behaviour is with both newly composed emails and replies.
I've noticed that the first 'stuck' email doesn't have a date or email icon, but looks like a draft email (i.e. no date, draft email icon).