The word 大忙{おおいそが}し is classified as a noun or a na-adjective.
If I analyze it correctly, the word 大忙し consists of the prefix (接頭語) "大" and the i-adjective stem of "忙しい." In addition, there is no such i-adjective as "大忙しい."
Out of curiosity, I'd like to ask if anyone knows other compound words that end with an i-adjective stem and are only used as a noun or a na-adjective (its i-adjective form doesn't exist).