Origins of many-particle interactions

The pair interaction is only a convenient approximation. More detailed approximations need multiparticle interactions. The most well-known one is the Axilrod-Teller 3-body potential which gives the first corrections to the $r^{-6}$ pair interaction describing for small distances the van der Waals repulsion of neutral atoms.

Another common way to model multiparticle interactions is the embedded atom model. Here the multiparticle interaction is not separable into k-body terms.

In general, a real system has some N-particle potential, which for theoretical reasons is translation, rotation, and permutation invariant. Theory gives no other restrictions. Thus which specific potential is correct must be decided by experiment (or deduced from a more detailed model). Pair potentials are simply the simplest class of approximations.

I'm not sure whether you're asking for examples of interactions where three particles interact simultaneously, or for systems where pairwise interactions are not adequate to describe the dynamics. So I'll offer you some of both!

Three-particle systems

You're made of the product of a three-particle interaction: the triple-$\alpha$ process, where three helium nuclei fuse to form carbon. The intermediate state, beryllium-8, has a lifetime around 10^-17 seconds. This is long by nuclear physics standards — enough time for light to cross the Be-8 nucleus 10⁸ times — but short enough that it takes roughly a solar mass of hot, electron-degenerate helium before there is enough beryllium for carbon production to take place at any realistic rate. This is why essentially no nuclei heavy than mass 7 were synthesized during the big bang. (And if it weren't for the coincidence that carbon has an excited state right near the energy of a ⁸Be and ⁴He together at rest, there might be no carbon in the universe at all.) But this is fundamentally a sequence of pairwise interactions, just taking place on very different timescales.

In a time-reversed world, there is recent evidence that iron-45, on the proton drip line, decays to chromium-43 by emitting two protons at once, rather than emitting the two protons sequentially or a bound diproton which decays later.

Of course, most weak decays have three particles in the final state (the daughter nucleus, the beta particle, and the neutrino); however the high-energy theory of the weak force describes this as a sequence of two two-body decays.

Three-body forces

Usually we describe what's happening inside a nucleus in terms of protons and neutrons. However, protons and neutrons are just the lowest-energy bound states of quantum chromodynamics; at very short distance scales, nucleons are made of many constituents, which interact strongly with each other where the nucleons overlap. In precision modeling of nuclei, one finds no pairwise interactions which describe all of the light nuclei. However there are several successful nuclear potentials which contain two- and three-nucleon forces.

You can compare the three-body force in nuclear physics to tidal forces arising in gravitation. The earth's tidal bulges arise because people with the moon overhead fall towards the moon faster than the rest of the earth does, while people with the moon underfoot feel the earth falling towards the moon away from them. This is a straightforward effect of the purely two-body gravitational force, which has major impacts on the long-term evolution of two-body systems (for instance, the moon's motion away from the earth as the pair become tidally locked, or the destruction of satellites within the Roche limit), and which vanishes if you use the otherwise-useful approximation that the objects involved are point masses. You could say in this case that gravitational tides are a "three-body" force that arises because there internal degrees of freedom to the earth-moon system that you'd prefer to suppress in your model.

The nuclear three-body force is the same sort of thing — except that unlike the tides, we don't actually have a quantitative theory for what those internal degrees of freedom are. I suspect that this is also what's happening in the interatomic potentials mentioned by Arnold. A complicated object like a nucleon or an atom may simply be a little different when it has close neighbors than when it's alone, and the sum of a two- and three- (and higher-)body interactions is a compact was to account for the difference.

You start your question with a fundamental statement from mechanics, and end with an example from a very complex system. Newtonian mechanics fundamentally involves only pair-wise forces at a microscopic level. When you build up a more complex system, like that of a catalyzed chemical reaction, you hide the very complicated microscopic phenomena by introducing higher levels of abstraction. For example, we don't discuss the individual collisions of a gas against a piston, instead we discuss pressure.

I seem to recall that someone (Mach?) has shown that interactions that require three objects (triple-wise potentials?) break Newtonian mechanics. That's just a hazy memory though. Furthermore, a three-body potential that is not the sum of pair-wise potentials would break superposition of forces. In any event, restricting attention to (microscopic) pair-wise potentials seems to work in all cases I'm aware of. @ArnoldNeumaier in his answer suggests that N-wise potentials work. I'd like to be corrected if that is true.