Is it permissible to take back one's wife after divorcing her twice—the first time with a single pronouncement, and the second time with three pronouncements—and to suffice with saying, "I have taken you back," without a new marriage contract?

If you pronounced your wife's divorce three times, intending for the three to take effect, then a major irrevocable divorce (baynunah kubra) has occurred, according to the majority of scholars. You are not permitted to remarry her unless she marries someone else, consummates the marriage, and then he separates from her, and her waiting period ('iddah) ends. However, if you repeated the divorce pronouncement merely for emphasis, then only one divorce occurs. It is advisable to present the matter to the Sharia court or to trustworthy scholars. If the right to return is established, do not disregard the document issued by the court to avoid harm.