We need to remember that an annulment does not invalidate a once valid marriage. An annulment is a decree that a valid marriage never existed. The short answers is, **yes **he would need a decree of nullity for *each *marriage in order to be free to marry.
If he is a Catholic, he was obligated to be married in the Catholic Church or receive permission to be married outside of it. If he is a Catholic and failed to receive permission to be married outside of the Church (or failed to have his marriage later validated in the Church) then the marriages are invalid. He would need to provide his Catholic baptismal certificate and copies of the civil marriages and divorces. Each marriage would need to be declared null.
If he was not a Catholic (or had permission to be married outside the Church or had his marriages validated in the Church), then his exchange of vows is a valid marriage. The first time he exchanged nuptial vows is the presumed valid marriage at this point in time. If the tribunal examines the marriage and finds that it was invalid, that would mean he was truly free to marry at the second time he exchanged marital vows. That marriage would become the presumed valid marriage unless proven otherwise. That marriage would then need to be examined by the tribunal. If it is determined to be invalid, then that means he was objectively free to marry the third time he exchanged marital vows and that becomes the presumed valid marriage. He would then need to have that marriage examined in order to determine its validity.
Only once all three prior exchanges of marital vows are determined to be invalid would he be free to enter into another attempt at marriage.