Are the times a husband divorced his wife—knowing that he had not consummated the marriage with her and each time he took her back then divorced her again—considered three divorces that prohibit him from taking her back, and is it permissible for him to take her back with a new contract and dower after her last divorce?

It appears that your husband divorced you with three pronouncements: The first was an irrevocable divorce (bainunah kubra) before consummation, and you are not required to observe an 'iddah (waiting period). A new marriage contract is necessary. The second pronouncement occurred after the second marriage contract and a valid sharia-compliant khalwa (private seclusion), which takes the place of consummation. This divorce is effective if the husband uttered it. Reconciling after this divorce is valid if it happened before the 'iddah expired, and the last pronouncement would then be the third. However, if the reconciliation occurred after the 'iddah, it is void, and a new marriage contract would be required. In this case, the khalwa would be unlawful, and any divorce pronounced after it would not be considered. If all three pronouncements of divorce are valid, then you are forbidden to this husband, and you will not become lawful for him unless you marry another man and he divorces you after consummation.