How many genders are there? As I understand it gender is a self-assigned identity and is correlated but not wholly determined by sex.

Sex is presumably determined by the sex chromosomes present in the DNA of the cells of the individual? The sexes are presumably male and female and intersex?

Is there a list of possible genders?
The subject of gender can be a very personal subject and the subject of gender is discussed among the LGBTQIA+ community.
The LGBTQIA Allyship is

the practice of confronting heterosexism, sexism, genderism, allosexism, and monosexism in oneself and others out of self-interest and a concern for the well being of lesbian, gay, bisexual, transgender, queer, intersex and asexual people.  Is founded on the belief and believes that dismantling heterosexism, monosexism, trans oppression/trans misogyny/cissexism and allosexism is a social justice issue (see the UCDavis link above)

The social sciences have a branch devoted to gender studies. Other sciences, such as sexology and neuroscience, are also interested in the subject.
The social sciences sometimes approach gender as a social construct, and gender studies particularly do, while research in the natural sciences investigates whether biological differences in males and females influence the development of gender in humans; both inform debate about how far biological differences influence the formation of gender identity. In some English literature, there is also a trichotomy between biological sex, psychological gender, and social gender role. This framework first appeared in a feminist paper on transsexualism (Yudkin, 1978).
Genders
There is the binary gender (male or female) and some identify themselves under a third (non-binary) gender of either:

non-binary,
genderqueer,
gender fluid,
bigender,
trigender,
pangender,
agender,
nongendered,
genderless,
genderfree, or
neutrois

See also: Gender Identity
Gender identity vs Sexual Orientation
Gender identity is separate from sexual or romantic orientation, and non-binary people have a variety of sexual orientations, just as transgender and cisgender people do.
A non-binary gender is not associated with a specific gender expression, such as androgyny. Non-binary people as a group have a wide variety of gender expressions. People who are non-binary have gender identities that are not exclusively masculine or feminine. They may identify as having an overlap of gender identities, having two or more genders, having no gender, having a fluctuating gender identity, or being third gender or other-gendered. Recognition of non-binary genders is still somewhat new to mainstream Western culture (McGee &amp; Warms, 2011).
Gender Identity and Society
Historically, many if not most societies have recognized only two distinct, broad classes of gender roles, a binary of masculine and feminine, largely corresponding to the biological sexes of male and female.

Most cultures currently construct their societies based on the understanding of gender binary—the two gender categorizations (male and female). Such societies divide their population based on biological sex assigned to individuals at birth to begin the process of gender socialization. (Nadal, 2017).

In addition to these traditionally recognized third genders, many cultures now recognize, to differing degrees, various non-binary gender identities.
Gender neutrality is a movement to end discrimination of gender altogether in society through means of gender-neutral language, the end of sex segregation, and other means.
Quantifying genders
Genders can only be quantified to a certain extent.  As indicated, there are binary and non-binary genders.  Binary being male or female and non-binary being others which don’t follow the binary convention. So, you could say there are three genders, but that is oversimplified.  The genders in the third (non-binary) gender category is numerous, and can be bigger than the list I provided.  It is all down to how people identify themselves in the way of gender.
References
McGee, R. J. &amp; Warms, R. L. (2011). Anthropological Theory: An Introductory History. New York: McGraw Hill.
Nadal, K. L. (2017). The SAGE Encyclopedia of Psychology and Gender
Yudkin, M. (1978). Transsexualism and women: A critical perspective. Feminist Studies. 4(3): 97–106. doi:10.2307/3177542 jstor: 3177542