Does the exact string theory $S$-matrix describe all physics there is? Suppose someone manages to evaluate the string theory $S$-matrix to all orders for any and all vertex operator insertions including non-perturbative contributions from world-sheet instantons and re-sum the whole series to obtain the exact non-perturbative string theory $S$-matrix for any combination of in- and out-states. Suppose further that the analytic result is compact, tractable, and easily amenable to numerical evaluations (say, some special function). Would such a result tell us "what string theory is"? Would it be enough in principle to answer all sensible questions about physics described by string theory? If not, what else is there we should care about?
 A: Well, for starters, the scattering matrix picture of interactions does not include the dynamics of spacetime, it is instead assumed as a background space where everything happens
Even string theory is just classical general relativity in a more fair description such that it can be quantized in a way that gives finite results for measurable quantities: it assumes that string modes contribute to $T_{\mu \nu}$ and as such, produce a curvature. The curvature of coherent excitations of a closed string has been proved to be equivalent to a small perturbation of the metric (see this question for details) and this gives string theorists confidence that such excitations describe gravitons.
But the picture of space-time is still classical, and a proper nonperturbative formulation of quantum spacetime is a revolution that still has not happened. Until that happens, no scattering matrix description can hope to be complete
A: No, first because string theory is based in a number of approximations/assumptions and second because not every physical question can be answered assuming that processes take an infinite time and involve objects separated infinitely as is assumed in the S-matrix approach.
The S-matrix approach is excellent for particle physics, which deals with few particles (usually two or three) in a large mostly empty volume and only considers initial and final states of free particles. The S-matrix approach fails when you start to study many-body motion in condensed phases. This is the reason why chemists have developed other theories beyond the S-matrix formalism for the study of chemical reactions, for instance.
