The Star Trek:
Enterprise episode "Cogenitor" introduces a race called the Vissians who have 3 genders: male, female, and cogenitor : The cogenitor does not pass on genetic material to the offspring they help create; Dr. Phlox suggested that they may supply an enzyme during the sex act which facilitates conception.
scifi.stackexchange.com/users/2287/gnovice
Isaac Asimov's The Gods Themselves has a three-gendered race.
Raphael Carter's short story "Congenital Agenesis of Gender Ideation" concerns the revelation of a number of different human genders.
A theatre troupe including SF author Geoff Ryman presented an improvised piece at the 1991 Edinburgh Festival called USEXCO that concerned the creation and marketing of two additional human genders.
scifi.stackexchange.com/users/23/Mike Scott
They show up from time to time.
Ian Banks' The Player of Games features a three gendered pan-human race.
Niven's Puppeteers are likewise three gendered (though one could argue about this as the bearing gender is strictly a separate species).
Le Guin's The Left Hand of Darkness features a human race that lives as neuters except during brief periods during which they take on one or the other gender (but not always the same one), which could be construed as three genders if you wanted to.
scifi.stackexchange.com/users/231/dmckee --- ex-moderator kitten
In the Alien Nation series the Newcomers have three genders.
Come to think of it, more than two sexes is not very plausible.
Finding a suitable mate is challenging enough in a two-sex species.
Imagine having to find two suitable mates for a three-sex species, or generally n-1 suitable mates for an n -sex species.
Your chances of reproducing will decrease rapidly as n increases.
scifi.stackexchange.com/users/638/Dima
The concept of a third sex is as old as writing.
In Greek myth there's the figure of Hermaphroditus .
In the Sumerian creation myth , the goddess Ninmah creates an intersex human from clay: Sixth, she fashioned one with neither penis nor vagina on its body.
Enki looked at the one with neither penis nor vagina on its body and gave it the name Nibru [perhaps = eunuch ], and decreed as its fate to stand before the king.
The idea recurs in other ancient cultures; see Wikipedia's third gender article for many more examples.
A few notable early examples in science fiction:
In A Voyage to Arcturus by David Lindsay (1920)
some of the characters belong to a third sex:
Nowadays there are men and women, but in the olden times the world was peopled by phaens .
I think I am the only survivor of all those beings who were then passing through Faceny's mind.
The Last Men in Olaf Stapledon's novel Last and First Men (1930) have many sexes, though these are apparently organized into "male" and "female" sexes: The traveller would recognize among us unmistakable sexual features, both of general proportions and special organs.
But it would take him long to discover that some of the most striking bodily and facial differences were due to differentiation of the two ancient sexes into many sub-sexes.
Full sexual experience involves for us a complicated relationship between individuals of all these types.
Of the extremely important sexual groups I shall speak again.
In
That Hideous Strength (1945) by C. S. Lewis
there's a cryptic reference to the "Seven Genders":
The three gods who had already met in the Blue Room were less unlike humanity than the two whom they still awaited.
In Viritrilbia and Venus and Malacandra were represented those two of the Seven Genders which bear a certain analogy to the biological sexes and can therefore be in some measure understood by men.
It would not be so with those who were now preparing to descend.
These also doubtless had their genders, but we have no clue to them.
These would be mightier energies: ancient eldils, steersmen of giant worlds which have never from the beginning been subdued into the sweet humilations of organic life.
Lewis didn't expand upon this, perhaps because he believed that it would be impossible to convincingly imagine a new sex.
In a 1943 letter he wrote, "Try to imagine a new primary color , a third sex , a fourth dimension , or even a monster that does not consist of existing animals stuck together .
Nothing happens."
(This now seems rather defeatist.
Maybe he should have tried harder.)
scifi.stackexchange.com/users/2502/Gareth Rees
Just to inject a bit of science into this: Sex is always well worth its two-fold cost claims to demonstrate why more than 2 sexes are evolutionarily unstable.
Why two sexes? claims to explain why two sexes has arisen so often in Terran evolution.
scifi.stackexchange.com/users/2945/Bruce Ediger
The existence of a three-gendered animal species actually forms a plot twist in
The Hugo and Nebula winning novel Dreamsnake by Vonda N. McIntyre
scifi.stackexchange.com/users/1459/Christi
How has this issue addressed? How was reproduction capable between the 3 or more genders?  What are the first instances where a race with more than 2 genders occurred?  I know of instances when there was 1 gender. A  neuter race  in Star Trek where the species chose to remove the sexuality from themselves.   I haven't however, seen any times when they try to introduce the concept of more than 2.
What is the history of the concept of more than 2 genders?
scifi.stackexchange.com
<races>
scifi.stackexchange.com/questions/5670
scifi.stackexchange.com/users/1109/OghmaOsiris
Enterprise  episode  "Cogenitor"  introduces a race called the  Vissians  who have 3 genders: male, female, and  cogenitor. The cogenitor does not pass on genetic material to the offspring they help create. Dr. Phlox suggested that they may supply an enzyme during the sex act which facilitates conception.   There are a number of examples of early work that dipicts the idea of races with more than 2 genders such as David Lindsay's " A Voyage to Arcturus". Star tracks episode "Cogenitor" depicts the idea and also shows how such as race re-produces and sustains itself. 
A primary example of 3 gender race that shows how it could exist and be produced lies in start trek episode "Cogenitor". The Cogenitor insert additional enzyms during reproduction in order to produce more genders within their race. There are a fewe early instances of races with more than 2 genders going as far back to touch on old Greek myths i.e. Hermaphroditus. Other examples include "A voyage to Arcturus" and "Last and First Men".