Fields have positive mass dimension, and local $n$-body interactions are suppressed by higher powers of the UV scale.
Indeed, in relativistic field theories in four dimensions the only renormalizable local 2-body interaction are scalar $\phi^4$ interactions, and the $A_\mu^4$ term hidden in the non-abelian Yang-Mills action. All other terms are local interactions of matter fields with Yang-Mills fields or scalars that lead to finite rage $n$-body interactions.
Note that non-abelian Yang-Mills fields not only lead to a Coulomb-like two-body interaction, but also to three and four-body forces (due to three and four gluon vertices). Indeed, at higher order in perturbation theory there are also higher n-body forces.
In non-relativistic theories the typical long range forces is mediated by a $U(1)$ gauge boson, which does not directly mediate n-body forces (although loop effects can mediate n-body forces). Finite range forces are subject to the power counting argument mentioned above. A standard example is nuclear physics, where the two-body interaction between neutrons and protons dominates, but three and four-body forces exist.
A beautiful non-relativistic system with some relevance to nuclear physics that can now be studied in cold atoms is the Efimov effect: A two-body force tuned to infinite scattering length (in a systems of bosons, or fermions with at least three degrees of freedom) requires a three-body force to be renormalizable.