  We study the highest states in the compact rank-1 sectors of the AdS5 X S5
superstring in the framework of the recently proposed light cone Bethe Ansatz
equations. In the su(1|1) sector we present strong coupling expansions in the
two limits L,lambda -> OO (expanding in power of lambda^{-1/4} with fixed large
L) and lambda, L -> OO (expanding in power of 1/L with fixed large lambda)
where lambda is the 't Hooft coupling and L is the number of Bethe momenta. The
two limits do not commute apart from the leading term which reproduces the
result obtained with the Arutyunov-Frolov-Staudacher phase in the lambda, L ->
OO limit. In the su(2) sector we perform the strong coupling expansions in the
L->OO limit up to O(lambda^{-1/4}), and our result is in agreement with
previuos String Bethe Ansatz analysis.
