Information that can be extracted from the time-ordered correlation function The time-ordered correlation function can be very complicated and encodes a tremendous amount of information. For example, the LSZ formula can be used to extract S-matrix elements from the time-ordered correlation function. What other quantities in the context of quantum field theory can be extracted from the time-ordered correlation function?
 A: In principle, all the information of the QFT is encoded in the $n$-point functions. This means that, once you know the correlation functions, you know the Hilbert space, the fields and their algebra (modulo a  unitary transformation). This is known as the Wightman reconstruction theorem, though AFAIK, it is not known whether the theorem holds for Yang-Mills theories (the Standard Model) or not: it is an open problem.
The details of the theorem are quite involved (and I'm not familiar with them, nor with AQFT in general), but if you want to read about Wightman's QFT this scholarpedia article seems nice.
Now I'll try to be more specific: in practice, what do we know once we know the $n$-point functions? The first thing, as you already noted, is that the $n$-point functions give us the information of all scattering phenomena. But apart from this, the $n$-point functions carry the information of the possible decays (and their time constants); the  poles of the correlators give you the energy of bound states (e.g., you can use this to calculate the mass of the proton using the QCD Lagrangian); the correlators are easily related to effective vertices (which contain the information of the electric and magnetic moments, for example); etc.
