What is the Sharīʿah ruling concerning the return of a couple after the husband has divorced his wife three times, knowing that the wife is a foreign convert to Islam, and the man married her with two legitimate Islamic contracts, and that she was taken back after the first and second times, while the third divorce occurred three days ago?

If the anger did not make you lose awareness of anything or control over your actions, and you divorced your wife three times, then she is not lawful for you until she marries another husband. Renewing the second marriage contract does not invalidate the previous divorce, by consensus. Your returning to your wife after the second divorce is valid if it was before the end of her waiting period (iddah) and does not require witnesses. If she returned before the end of her waiting period, then she has become irrevocably divorced from you (bainunah kubra) if anger did not overcome your intellect, and she is not lawful for you until she marries another husband. It is forbidden for you to live with this woman after her irrevocable divorce, as she is a stranger to you.