He married her in a ‘urfi marriage, then he divorced her three times. Is it permissible for him to marry her after that in a valid marriage without her first having been married to another husband?
If what you mean by ‘urfi marriage is that you married the woman in a valid, shar‘i marriage with a mahr, and in the presence of witnesses and the woman’s wali (guardian), but you did not announce it to people or have it documented in the court, and you concealed it from your first wife, then it is a valid marriage according to Islam, and divorce, if it happens, is valid beyond any doubt. So it is not permissible for you to (re)marry her after having divorced her three times, until she has been married to another man in a genuine marriage, not a tahleel marriage (one aimed at making it permissible for her to go back to her first husband). The Prophet (blessings and peace of Allah be upon him) cursed the muhallil and the muhallal lahu. [The muhallil is the one who marries a woman and divorces her so that she can go back to her first husband, and the muhallal lahu is the first husband.]
For more information on the invalidity of the tahleel marriage and the prohibition thereon, please see the answer to question no. 109245.
If what you meant by ‘urfi marriage is that you married her without the permission of her guardian and did not tell anyone because you were afraid that the first wife would find out, then there is a difference of opinion among the scholars as to whether a woman’s marriage without the permission of her guardian is valid. Imam Abu Haneefah (may Allah have mercy on him) was of the view that it is valid, but the majority of imams are of the view that it is not valid, based on a great deal of evidence, including the words of the Prophet (blessings and peace of Allah be upon him) “Any woman who gets married without the permission of her guardian, her marriage is invalid, her marriage is invalid, her marriage is invalid.”
Narrated and classed as hasan by at-Tirmidhi, 1102; Abu Dawood, 2083; Ibn Maajah, 1879; from the hadeeth of ‘Aa’ishah (may Allah be pleased with her). Classed as saheeh by al-Albaani in Irwa’ al-Ghaleel, 1840.
According to both opinions (the view of Abu Haneefah and the view of the majority), so long as you married her believing that the marriage was valid, the rulings on valid marriage are binding upon you, so your divorce counts as such.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
Divorce counts as such in a marriage concerning the validity of which there is a difference of opinion, if he believed that it was valid. End quote.
Majmoo‘ al-Fataawa, 32/99
It is not permissible for a man, if he has divorced his wife three times, to ask about the validity of his marriage and seek to declare it and the three divorces he issued as invalid, because the one who is toying with the rulings of Allah regard the marriage as valid when it suits him and invalid when it suits him.
We have previously quoted the opinions of a number of scholars concerning this issue, in the answer to question no. 116575
Based on that, it is not permissible for you to (re)marry this woman until she has been married to another husband.
And Allah knows best.