By functional, I mean in the reproductive sense. While many different male fish fertilize multiple female fish's eggs, and flowers pollinate chaotically at one another, are there any species that require three or more sexes to successfully breed? For example, say one bird fertilized another bird's egg but needed a third sex with an organ for incubation.
Yes, but it doesn't work the way you think. Slime molds have quite a few sexes (13 or over 500, depending on how you count), but any two can breed, as long as they're different sexes.
So yes, there are multiple sexes, but it only takes two to tango.
Additionally there are fungi that have several thousand.
The reason is that a `child' can only mate with a small fraction of siblings but with a high fraction of non-siblings, thus leading to greater genetic diversity via sexual selection (I'm a computer programmer who worked in a Biolab, so please excuse me if I am slightly off and abuse terminology).