  We consider the exact coupling constant dependence of extremal correlation
functions of ${\cal N} = 2$ chiral primary operators in 4d ${\cal N} = 2$
superconformal gauge theories with gauge group SU(N) and N_f=2N massless
fundamental hypermultiplets. The 2- and 3-point functions, viewed as functions
of the exactly marginal coupling constant and theta angle, obey the tt*
equations. In the case at hand, the tt* equations form a set of complicated
non-linear coupled matrix equations. We point out that there is an ad hoc
self-consistent ansatz that reduces this set of partial differential equations
to a sequence of decoupled semi-infinite Toda chains, similar to the one
encountered previously in the special case of SU(2) gauge group. This ansatz
requires a surprising new non-renormalization theorem in ${\cal N} = 2$
superconformal field theories. We derive a general 3-loop perturbative formula
for 2- and 3-point functions in the ${\cal N} = 2$ chiral ring of the SU(N)
theory, and in all explicitly computed examples we find agreement with the tt*
equations, as well as the above-mentioned ansatz. This is suggestive evidence
for an interesting non-perturbative conjecture about the structure of the
${\cal N} = 2$ chiral ring in this class of theories. We discuss several
implications of this conjecture. For example, it implies that the holonomy of
the vector bundles of chiral primaries over the superconformal manifold is
reducible. It also implies that a specific subset of extremal correlation
functions can be computed in the SU(N) theory using information solely from the
S^4 partition function of the theory obtained by supersymmetric localization.
