Can a Qatari man who divorced his wife three times without having witnesses for each divorce return to her without her marrying another man then divorcing him?

Divorce takes effect the moment the husband pronounces it, and witnesses are not a condition. If you pronounced the three divorces separately during different periods of purity, and you took your wife back after each pronouncement, then she is irrevocably divorced from you with a major separation (baynunah kubra). She is not lawful for you until she marries another husband in a desired marriage and he divorces her after consummating the marriage. However, if any of the aforementioned conditions are not met, she is also irrevocably divorced from you with a major separation, according to the majority of scholars, and this is the most preponderant opinion.