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 Qur’aanic 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 Allaah

(peace and blessings of Allaah be upon him) and said, “O Messenger of
Allaah, 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 Allaah

(peace and blessings of Allaah 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-Bukhaari, 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 Allaah knows bets. May Allaah bless our Prophet Muhammad.