The multiple expansion of a potential V has contributing terms proportional to $\frac{1}{r^{n+1}}$ where $n=0,1,2...$.
First, why are we interested only in integer powers of r?
Second, why are we interested in symmetry pole arrangements? In other words, the $1/r$ term is a monopole, the $1/r^2$ term is a dipole, the $1/r^3$ term a quadrupole -- all of these are symmetric configurations. Why do we not consider an asymmetric 'tripole' formed in an isosceles triangle?
The reason I pose these questions is that my understanding of this formalism is that it represents the potential far away from an arbitrarily complicated charge configuration. But if the charge configuration is arbitrarily complicated, then why do we assume such perfectly symmetric poles?