  We prove that Calabi-Yau metrics on compact Calabi-Yau manifolds whose Kahler
classes shrink the fibers of a holomorphic fibration have a priori estimates of
all orders away from the singular fibers. To this end we prove an asymptotic
expansion of these metrics in terms of powers of the fiber diameter, with k-th
order remainders that satisfy uniform C^k-estimates with respect to a
collapsing family of background metrics. The constants in these estimates are
uniform not only in the sense that they are independent of the fiber diameter,
but also in the sense that they only depend on the constant in the estimate for
k=0 known from previous work of the second-named author. For k>0 the new
estimates are proved by blowup and contradiction, and each additional term of
the expansion arises as the obstruction to proving a uniform bound on one
additional derivative of the remainder.
