Is there an electronic structure method which accounts for 3-, 4-, ..., n-electron exchange integrals?
I am posting this answer not as the definitively correct answer, but as an example of the type of thing I am talking about.

azago's answer is correct in pointing out that the Slater determinant contains within it the correct permutational symmetry of the wavefunction required by electrons. Namely, it is antisymmetric upon interchange of two orbitals. Additionally, it symmetric upon interchange of three (or any odd number of) orbitals. This is the symmetry which must be present in higher cycles of exchanges.

What I have found upon further reading is that I could have been more precise in the wording of my question. Namely, these higher-order exchange interactions are not surprising if I had made it clear that these are necessarily three-center, four-center,... integrals. Then it is more clear that HF, while having the correct symmetry to adhere to Pauli exclusion, does not include these integrals as it only deals with 2-electron integrals at the most.

So, I will discuss three contexts I have found in which the types of integrals I am thinking of show up.

Hamiltonian for a Three-Atom System:

Following along with ref. [1], the Hamiltonian for the three atom system, $\ce{H3}$ is,
$$
H=\sum_i\left(-\frac{1}{2}\nabla^2_i-\sum_{\alpha}\frac{1}{r_{\alpha i}}+\sum_{j\gt i}\frac{1}{r_{ij}} \right)+\sum_k\frac{1}{R_k}
$$
where $i$ and $j$ are labels for the three electrons, and $\alpha$ is the label for each Hydrogen atom with locations $R_1$, $R_2$, and $R_3$.

It is clear from this Hamiltonian that we have the usual coulomb terms and the usual two-electron integrals. Something weird is going to happen with the exchange terms though. There are actually three possible configurations of the atoms, but the only point I'm making here is that the matrix elements for one of the atomic configurations can be written as,
$$
\langle\psi_1|H|\psi_1\rangle=4J+4K_1-2K_2-2K_3-4K_{123}
$$
I have deviated from their notation here for consistency with current convention. These integrals are written explicitly as:
$$
\begin{align}
J&amp;=\langle abc|H|abc\rangle \\
K_1&amp;=\langle abc|H|bac\rangle \\
K_2&amp;=\langle abc|H|acb\rangle \\
K_3&amp;=\langle abc|H|cba\rangle \\
K_{123}&amp;=\langle abc|H|cab\rangle \\
\end{align}
$$
where now it is quite clear that $K_{123}$ is a double exchange. That is, two electron labels are changed, and it is clear this is distinct from the other exchange integrals which are the usual exchanges we are used to.

I do not think this has anything to do with the fact there are three atoms present, however. Only that there are three electrons. Thus, I believe that even a complete description of Lithium would include these three-electron exchanges. Thus, the reason we do not see these integrals in HF is that they are left out of the Hamiltonian.

Explicitly-Correlated Methods:

I'll be brief here because explicitly-correlated methods are pretty complicated and I haven't spent enough time studying them to really know what's going on, but one thing which does happen is that you get matrix elements of the following form[2]:
$$
Z_{\nu w,p}^{xy,q}=\langle w_{\nu w,p}|(1-\hat{p}_{13})r_{13}^{-1}+(1-\hat{p}_{23})r_{23}^{-1}|w_{xy,q}\rangle
$$
The only important in this horrible integral is that the exchange operator $\hat{p}_{ij}$ appears with the labels of three-distinct electrons. Now, I have no idea if it's valid to interpret this as the type of integral I'm talking about, but these kinds of integrals appear all over the place in explicitly correlated methods.

Three-Body Symmetry-Adapted Perturbation Theory

Now, two-body symmetry-adapted perturbation theory (SAPT) is quite a common method, but there is a 3-body version of SAPT which can be quite a useful tool at times.

In Ref. [3], the authors present the relevant equations for 3-body SAPT, but they note that they approximate the exchange term out to the fourth power of orbital overlap. That is, for the exchange operator they do the following:
$$
P\approx 1+P_2+P_3+P_4
$$
They describe this as follows:


  In this expression, $P_2$ represents the operator which interchanges
  one electron from $X$ with one electron from $Y$ and
  therefore produces terms proportional to $S^2$. The operator
  $P_3$ includes double exchanges between three monomers involving
  three electrons. This operator, absent for a dimer,
  generates terms proportional to $S^3$. The operator $P_4$ is a
  double exchange operator which interchanges four electrons
  among the monomers and generates terms proportional to
  $S^4$. The double exchanges produced by $P_4$ are of the
  $(XY)(XY)$ or $(XY)(XZ)$ types and can be thought of as two
  exchanges of the type produced by $P_2$ occurring simultaneously.
  In contrast, double exchanges of the $(XY)(XYZ)$
  type are included in $P_5$ and double exchanges of the
  $(XYZ)(XYZ)$ type are included in $P_6$, leading to terms proportional
  to $S^5$ and $S^6$, respectively.


The whole purpose of three-body SAPT is to be able to describe the 3-body non-additivities which arise from intramolecular interactions. It has already been pointed out that the electrostatic interaction is additive up to 2-body because Coulomb's law follows the superposition principle. This then makes it clear that these higher-order exchanges involving more than two electrons generate some of the non-additive forms of coulomb correlation.

Finally, the authors of ref. [3] provide the form of these three-electron cycles I was trying to describe which can be written as:
$$
P_{111}=\sum{i\in A}\sum{i\in B}\sum{i\in C}[P_{ij}P_{jk}+P_{ik}P_{jk}]
$$
which interchanges electrons in a cycle on atoms $A$, $B$, and $C$. This operator generates all possible three-electron exchanges between three atoms. There are no intra-monomer exchanges however.



[1] Porter, R. N., &amp; Karplus, M. (1964). Potential energy surface for H3. The Journal of Chemical Physics, 40(4), 1105-1115.

[2] Hättig, C., Klopper, W., Köhn, A., &amp; Tew, D. P. (2011). Explicitly correlated electrons in molecules. Chemical reviews, 112(1), 4-74.

[3] Lotrich, V. F., &amp; Szalewicz, K. (1997). Symmetry-adapted perturbation theory of three-body nonadditivity of intermolecular interaction energy. The Journal of chemical physics, 106(23), 9668-9687.