Why is it assumed supersymmetry in nature is ${\cal N}=1$? It is often presumed that if supersymmetry exists then there will be a single supersymmetry (${\cal N}=1$). Why do we assume this? What is wrong with additional supersymmetries (${\cal N}>1$)? For example 4D supergravity with ${\cal N}=8$?
 A: It is because the Standard Model is a chiral theory, i.e. there are particles whose components of different chirality transform diferently under gauge symmetry. Only $N=1$ allows for chiral matter since it can accommodate right and left fermions in chiral multiplets.
For $N=2$ we can accommodate fermions in vector multiplets and hypermultiplets, but the former shall transform in the adjoint of the gauge group whereas the latter contain both chiralities. Therefore this does not work. Similarly, $N=4$ cannot have fermions transforming in the fundamental representation of the gauge group. For this reason, extended supersymmetry must be broken down to $N=1$.
A: I think what you are asking is extended supersymmetry. That is when N>1. it is when you have more then one kind of SUSY transformation.
The more SUSY the theory has, the more constraint the field and interactions.
In 4D, spinor has 4 degrees of freedom and so there are 32 generators.
In this case, where N=8, there is a graviton in the theory and it is called supergravity.
In this case you actually have to reduce the number of dimensions from 11 to 4 by setting the size of 7 dimensions to 0.
It has 8 SUSYs, and that is the max since you go half steps from -2 to 2 spin.
Spin 2 gravitons are the particles with the most spins.
Your question was why N=1, for model building it is usually assumed that almost all SUSYs would be broken in nature, leaving just 1 SUSY, although nowadays, there is lack of evidence for N=1 so N=2 and higher are considered.
