Is it Islamically permissible for a husband to divorce his ex-wife a second and third time after the expiration of her iddah from the first divorce, and without him taking her back into his responsibility, due to her need for this divorce?

A revocable divorcee becomes irrevocably divorced upon the expiration of her waiting period (iddah), at which point the marital bond between the spouses is severed. Therefore, no divorce can be issued to her after her irrevocable separation, because it would not be directed at a valid object. Thus, it is not permissible for a man to divorce his wife the remainder of the three divorces after her irrevocable separation and the expiration of her waiting period, with the caveat that if the divorce exceeds one pronouncement without a resumption of conjugal relations between each two pronouncements, it constitutes an innovated divorce (talaq bid'i), which is impermissible to issue.