The novel "The Player of games" by Iain M. Banks describes an alien society with 3 sexes. They are otherwise pretty humanoid, but apart from male and female, there is an intermediate gender called "apex". Reproduction requires the male to fertilize the egg carried by the apex, then the apex planting the fertilized egg in the female's womb. This led to the discrimination of women based on the belief that they did not really contribute to reproduction, although later research showed that the females do alter the foetal DNA via a retrovirus.
The society formed by these people is a pretty ancient interstellar empire dominated by the apexes. Males and females are treated more like slaves or property - selective breeding and genetic engineering actually led to them being less intelligent than apexes on average. Because of their greater physical strength, males serve as labourers and form the bulk of the military (at least the lower ranks). Females, at least those of higher social standing (meaning basically that they belong to the household of an influential an apex), are treated as valuable property.
So this raises the frightening question what happens if a society manages to make certain cliches/gender roles "come true" - or what if they naturally are?. Although this is not dependent on the number of sexes, it could be emphasized in a multi-gender society as there is more room for specialized roles. E.g. you could also claim that bees have 3 "sexes" (the queen as female, the drones as male and the workers who are technically female but do not breed as "neutral" sex). In this example, you don't even have all sexes participating in reproduction. A race modelled on the concept of an insect hive could have vast differences in capabilities, intelligence and roles (see Formians in the MM for a D&D example).
