Are there lifeforms that have more than 2 sexes?
There exists something like multiple sexes in fungi, but it's a bit complicated.

First of all fungi don't actually have anything that represents the two classical animal sexes, phenotypically their reproductive cells are all equal (they have isogamic reproduction).
However there can be certain alleles in some fungi species (sometimes multigenic, sometimes not), that restrict sexual reproduction to other fungi with a different variant of that allele, these alleles are therefore called mating types. It is however important to note, that almost all fungi can also reproduce asexually (by basically cloning themselves).
In general all of the mating type systems always require only 2 individuals to (sexually) reproduce, it just can't be 2 individuals with the same type.

In their simplest form mating types resemble the male/female sex we often see in animals: for some fungi (e.g. yeast) there exist exactly 2 forms of the mating type allele (called a and $\alpha$), so basically they have '2 sexes' (though genetically/biologically the difference is MUCH smaller). 

There are however also much more complicated mating type systems: 
Some species of the Basidiomycota or Agaricomycotina groups have a tetrapolar mating type, which is composed of 2 independent alleles (with 2 variants per allele this would already results in 4 different 'sexes'). Additionally they can have many more than 2 forms for each of the alleles (up to hundreds), which leads to (ten-)thousands of different mating type combinations (or 'sexes').