  We compute the one-instanton effective action of N=4 super Yang-Mills theory
with gauge group Sp(2N). The result can be written in a very compact and
manifestly supersymmetric form involving an integral over the superspace of an
irrational function of the N=4 on-shell superfields. In the Coulomb branch, the
instanton corrects both the MHV and next-to-next-MHV higher derivative terms
D^4F^{2n+2} and F^{2n+4}. We confirm at the non-perturbative level the
non-renormalization theorems for MHV F^{2n+2} terms that are expected to
receive perturbative corrections only at n-loops. We compute also the one and
two-loop corrections to the D^4F^4 term and show that its completion under
SL(2,Z) duality is consistent with the one-instanton results of U(2) gauge
group.
