I wish Microsoft included a second checkbox for the Choice Column labeled "Capture 'Fill-in' choices" that would automatically add the user's fill-in choice to the dropdown list.
I'm using SP2013. I don't have access to Designer, or InfoPath, or SP Event Receivers, or Nintex Workflow.
If I make progress, I'll post it here too.
There appears to be some code HERE that explains how to retrieve the value in the fill-in text box. Just need to work out how to trigger, then update the drop-down list.