Some fungi have thousands of "sexes"
The answer to your question depends on why you mean by a sex. Male and female are dimorphic, and have separate roles in reproductive development that correspond at the fundamental level to micro- and macro- gametes. The female produces fewer, larger, less mobile gametes than contain more nourishment to get the development started; in contrast to the male producing many, small, typically mobile gametes. Everything about the distinction of sexes essentially follow from this starting point.
Fungi do it differently. There's no morphological distinction between the "sexes", or more properly, "mating types", instead cells of the two parents merge directly to form diploid cells that later divide. The importance of the mating types is simply to reduce inbreeding so two individuals of the same mating type can't interbreed. In some branches of the fungal tree species can have thousands of different mating types (i.e. BasisiomycetesBasidiomycetes).
So, the question of whether you can have more sexes depends on what roles you expect your sexes to take. Are you looking for a system in which >2 individuals are involved in each mating event? Do you require distinct morphology for your sexes?
The suggestions you put forward are implausible on the face of it - there must be individual advantage in every role for the system to persist. One possible way round this if the sexes rather than being fixed as in humans are life stages as in some fish (i.e. angler fish which start male and become female if they grow big enough) or if you imagine that the roles are more closely akin to castes in Eusocial species.