Can he take his wife back after her ‘iddah has ended?

If a man divorces his wife with the first or second talaaq, once her ‘iddah is over she is divorced and becomes a “stranger” (non-mahram) to him; she cannot go back to him except with a new marriage contract which fulfils all the conditions set out in sharee’ah (see Question no. 2127). 

But if he divorces his wife with the third talaaq, then she becomes haraam to her first husband until she has married a second husband in a genuine marriage which is consummated. The Quranic evidence for that is the aayah (interpretation of the meaning): 

“The divorce is twice, after that, either you retain her on reasonable terms or release her with kindness…

And if he has divorced her (the third time), then she is not lawful unto him thereafter until she has married another husband” [al-Baqarah 2:229-230]

 The last divorce refers to the third divorce, according to all the scholars. 

The evidence from the Sunnah is the hadeeth narrated by ‘Urwah ibn al-Zubayr, according to which ‘Aa’ishah told him that the wife of Rifaa’ah al-Qurazi came to the Messenger of Allah (peace and blessings of Allah be upon him) and said, “O Messenger of Allah, Rifaa’ah divorced me thrice, then I was married to ‘Abd al-Rahmaan ibn al-Zubayr al-Qurazi, but he has nothing with him except something like this fringe. The Messenger of Allah (peace and blessings of Allah be upon him) said, “Perhaps you want to go back to Rifaa’ah? No, (it is not possible), unless and until you enjoy the sexual relation with him (‘Abd al-Rahmaan), and he enjoys the sexual relation with you.”

(Narrated by al-Bukhari, 4856; Muslim, 2587) 

The phrase “divorced me three times” means that this was the divorce which meant that she was no longer married to him, which is the third talaaq. The Prophet’s words “until you enjoy the sexual relation with him (‘Abd al-Rahmaan), and he enjoys the sexual relation with you” refer to intercourse. 

Al-Nawawi said: “This hadeeth indicates that the woman who has been divorced by a third talaaq is not permissible to the man who has divorced her until she has been married by another husband, who has intercourse with her then separates from her, and she completes her ‘iddah. Merely drawing up the marriage contract with her does not make her permissible to the first husband. This is the view of all the scholars among the Sahaabah, Taabi’een and those who came after them.  Sharh Muslim, 10/3 

And Allah knows bets. May Allah bless our Prophet Muhammad.