Does divorce take effect if the husband utters "You are divorced, divorced, divorced" three consecutive times in a state of extreme anger, after the wife requested it?

A wife is not permitted to seek divorce without a legitimate Shari (Islamic legal) excuse. What your husband uttered, "You are divorced," is an explicit pronouncement of divorce and takes effect without the need for intention. If this was the third pronouncement of divorce, then you are unlawful for him, and you will not become lawful for him until you marry another husband and he consummates the marriage with you, and then he divorces you—unless he was in a state of extreme anger where he did not comprehend what he was saying. If your husband intended by repeating "divorced, divorced, divorced" to emphasize rather than to issue multiple divorces, then only one divorce takes effect, and he has the right to take you back.