Well this is disappointing

Multiple sources say the Chukchi people of Siberia have, or had, seven genders, citing a 1904 account by Russian anthropologist Waldemar Borgoras. For example, Sue-Ellen Jacobs and Christine Roberts in the chapter cited below say that "three were female and four were male". Looking at his account, I can only find four: man, "soft man" / "similar to a woman", woman, and "similar to a man". I'll pursue this - it'd be much more interesting if there really were three more.
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Borgoras, Waldemar. "The Chukchee". in Franz Boas (ed). The Jesup North Pacific Expedition. Memoir of the American Museum of Natural History 11, part 2, 1904, pp 449-451.

Jacobs, Sue-Ellen and Christine Roberts. "Sex, Sexuality, Gender, and Gender Variance". in Sandra Morgen (ed). Gender and Anthropology: critical reviews for research and teaching. Washington DC, American Anthropological Association, 1989. p 440.


ETA: I thought this might be the case: Jacobs and Roberts count the intermediate stages of gender transformation as separate genders. Women start by changing how their arrange their hair, then begin dressing as men, and finally take on men's behaviour and tasks; men follow the same pattern. The seventh gender in this schema is the transformed person "feeling like" their new sex and taking on an appropriate spouse and domestic role.
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Jacobs, Sue-Ellen and Jason Cromwell. Visions and Revisions of Reality: Reflections on Sex, Sexuality, Gender, and Gender-Variance. Journal of Homosexuality 23(4), 1992. pp 43-69.