Populate Multi choice options with user responses

Possible, but it won't be perfect.
You would have to allow for a lot of new choices...so say choices 1 - 100. Use display logic to only show if the value is not empty for an embedded data variable.
In survey flow (at the top) set the embedded data variables values from a contact list, pulling in values for choices 1 - 100. NOTE: you can only pull from one contact list per survey so if you are already using a contact list this won't work.
As people respond, and fill in the Other option. You would need to run through a branch in Survey Flow, ONLY if the Other value was not blank. You then would search for the first blank value in your choices. Once found, you'd fill that variable with the "Other" text entry. If you only anticipate 10 this is doable. if it keeps growing it could be more effort than worth the benefit.
I'd also caution you that you will have no control of what is added as a choice...so you could end up with inappropriate items being added to your list for your next respondent. ALSO anyone that has already started the survey, would not see the new items. IF you are doing this as a workflow, where the next respondent won't see it until the previous one finishes, this is much more doable.