  We consider leading strong coupling corrections to the energy of the lightest
massive string modes in AdS_5 x S^5, which should be dual to members of the
Konishi operator multiplet in N=4 SYM theory. This determines the general
structure of the strong-coupling expansion of the anomalous dimension of the
Konishi operator. We use 1-loop results for several semiclassical string states
to extract information about the leading coefficients in this expansion. Our
prediction is
  Delta= 2 lambda^{1/4} + b_0 + b_1 lambda^{-1/4} + b_3 lambda^{-3/4} +...,
where b_0 and b_1 are rational while b_3 is transcendental containing zeta(3).
Explicitly, we argue that b_0= Delta_0 - 4 (where Delta_0 is the canonical
dimension of the corresponding gauge-theory operator in the Konishi multiplet)
and b_1=1. Our conclusions are sensitive to few assumptions, implied by a
correspondence with flat-space expressions, on how to translate semiclassical
quantization results into predictions for the exact quantum string spectrum.
