  We calculate via the effective field theory (EFT) approach the
next-to-next-to-leading order (NNLO) spin1-spin2 conservative potential for a
binary. Hereby, we first demonstrate the ability of the EFT approach to go at
NNLO in post-Newtonian (PN) corrections from spin effects. The NNLO spin1-spin2
interaction is evaluated at fourth PN order for a binary of maximally rotating
compact objects. This sector includes contributions from diagrams, which are
not pure spin1-spin2 diagrams, as they contribute through the leading-order
spin accelerations and precessions, that should be first taken into account
here. The fact that the spin is derivative-coupled adds significantly to the
complexity of computations. In particular, for the irreducible two-loop
diagrams, which are the most complicated to evaluate in this sector,
irreducible two-loop tensor integrals up to order 4 are required. The EFT
calculation is carried out in terms of the nonrelativistic gravitational (NRG)
fields. However, not all of the benefits of the NRG fields apply to spin
interactions, as all possible diagram topologies are realized at each order of
$G$ included. Still, the NRG fields remain advantageous, and thus there was no
use of automated computations in this work. Our final result can be reduced,
and a corresponding Hamiltonian may be derived.
