Getting inspiration from electrostatics, where in the potential energy expressions monopole charge multiplies electric potential and dipole moment multiplies gradient of electric potential, we can extrapolate. For the order $n$ electric moment ($2^n$-pole), the number of spatial derivatives of electric potential in the potential energy expression should be $n$.
I.e. for monopole (moment of order $n=0$) we have potential energy $q\varphi$, for dipole (order $n=1$), we have $d_j \partial_j \varphi$, for quadrupole (order $n=2$), we have $\frac{1}{2}Q_{jj'}\partial_j\partial_{j'}\varphi$, and so on.
For magnetic moments this should be similar, except instead of electric potential, vector potential should be used, and also instead of using pseudovector for the moment, we should use antisymmetric tensor: for the usual magnetic moment vector $\mu_k$, we can define anti-symmetric tensor $M_{lm} = \mu_k \epsilon_{klm}$ to play the role of "the characteristic of the moment of first order that multiplies the first-order derivative of the potential". Then we have potential energy $-M_{lm}\partial_l A_m$. For the 2nd order magnetic moment, we can construct something like $-\frac{1}{2}M_{l'lm}\partial_{l'} \partial_{l} A_m$, and so on.
Of course this is just a plausible formal procedure of generating new ugly Hamiltonians for hypothetical systems that have important interaction of $n$-th order moment with the external field (like it happens with the Pauli term). Whether potential energy of a charge or current distribution in external field can be expressed correctly this way is a more involved question from EM theory, and the answer is probably yes, but the moment has to be properly defined (see the definition of the quadrupole moment in electrostatics that makes the energy formula valid, it's not something one could guess by extrapolating the number of multipling coordinates in the integral for dipole moment).
