How many fundamental constants does it take to describe our Universe?

The answer to this question has changed over time. My knowledge of the history is incomplete to say the least. During the first super string revolution, when it was understood that perturbative string theory would give a description of perturbative quantum gravity, the goal was to demonstrate that there were only a few choices of compactification manifolds available from consistency arguments (anomaly cancellation and unbroken super symmetry in four dimensions, probably the most well known paper is "Vacuum configurations for superstrings" by Candelas et. al.)

Ignoring heaps of details, the result was that the vacuum manifold (that is a solution to the 10d vacuum Einstein Equations) had to be some 4 dimensional symmetric space (usually taken to be Minkowsky Space) times a Calabi-Yau manifold, those are Kaehler manifolds with vanishing Ricci curvature (those have non-vanishing constant spinors, which you need for unbroken supersymmetry).

Initially it was believed that with additional physical consistency conditions and by relating the geometry of the Calabi-Yau manifold to known physical parameters, like the number of generations of particles, it would be possible to narrow down the number of "stable" vacua to a limited number.

While there were discovered a fair amount of dualities between different string theories, with the discovery of D-Branes and Flux Compactifications (which stabilize the moduli of a huge number of potential vacua, the general idea is that the space of calabi-yau manifolds can locally be paramterized as a finite dimensional manifold and those paramters would show up as massless fields that aren't observed, if there weren't any fluxes they couple to), the number of potentially stable vacua is now sometimes quoted as 10^500.

In itself this is not too concerning, if you think about the fact that there is a infinite number of solutions to the Einstein Field equations and general relativity is still a very predictive theory. In practice this means that you carefully engineer the vacua you study, for example with F-theory techniques in order to relate them to observed phenomena.

You are right that it just reframes the question, the low energy effective field coupling constants turn up as vacuum expectation values of certain fields, but in an interesting way. The idea is then to use relations and identities discovered in string theory (for example AdS-CFT) to study the low energy theories.

Works for creationists, though. Their constant is "God" who they then define as a number of infinities on various axes. Then they tell you that you're not allowed to question it.

String theory much like Gauge theory is not fine tuned to any set of observed phenomena and for the most they don't make numerical predictions. What they do predict is the general shape of cross-sections for different particle species and corrections to them in the case of Gauge theories and much in the same way one can work out various String scattering cross sections. In the case of Gauge theories this is how you determine which of them fits the measured cross section best. In the case of String theories this failed when it was still believed to be a theory of the strong interaction. It eventually was realized that the string scale lS is related to the Planck lP by

lP = lS * gS^{1/4}

where gS is the vacuum expectation value of the Dilaton. It is believed to be of order one, so lS is ~10^35 meters. The current accessible length scale at the LHC ~10^18 meters, so you have 17 orders of magnitude between them and in particular no reason to expect the string scattering cross sections to relate to anything we can observe directly.

What is found instead is that string scattering amplitudes assemble in a way that predict that the massless string excitations form Field theories already known, like type IIB supergravity, IIA supergravity, SO(32) and E(8) x E(8) coupled to supergravity and a few others. What is striking about those initial calculations, is that from field theory arguments those field theories are the only supersymmetric ones in 10 dimensions incorporating gravity.

Perturbative String theory in its current form makes most of the model input data part of the "vacuum geometry", much in the same way Einstein did with general relativity. String theory in addition has the advantage that it gives a consistent perturbative description of quantum gravity.

String phenomenology, that is the study of String theory with the purpose of making low energy predictions is still in its infancy. Its value at the moment is primarily in coming up with consistent extensions of the Standard Model, the alternative of conservative step-wise refinements (conjecture additional particles to exist and their coupling mechanism to the known particles, calculate cross sections and hope that they aren't ruled out by experiments yet, for example cause existing particles to decay that aren't really supposed to, like the proton) is still the majority approach. Dark matter and other theoretical considerations tell us however that the Standard Model is incomplete.

The attractiveness of the string theory approach is that in some ways it is more restrictive as you have to satisfy additional geometric constraints, whereas in standard gauge theory the only consistency constraints are unitarity and locality, which predicts that fundamental particles will have spin 0,1/2,1 and 2 and renormalizability which restricts you essentially to Yang-Mills theory, plus some extensions (Supersymmetry among others). What isn't fixed is the gauge group, the precise way Fermions couple to each other etc. Some of those can't be fixed because things like the coupling constants aren't actually constant but dependent on the energy scale (they are related by the renormalization group).

Most likely if we find some new particles at the LHC it will be possible to give them a low energy effective field theory description (maybe involving supersymmetry), if we don't find things like extra dimensions. In other words it is unlikely that we are forced to consider string theory in order to understand physics at the LHC scale, its value is that it potentially has better conceptual tools to come up with super-symmetric extensions of it.

I suppose his answer might be more philosophical. All things in this universe are shards of the same one universe. Shards of this one universe thinking about what this universe is itself made of. As far as we know from the fact the constants cannot be derived from any other principle and must be measured directly, the universe is made of itself. The universe is one.

Also an interesting read, if you're feeling philosophical about the number 1. http://en.wikipedia.org/wiki/One-electron_universe

EDIT: I'm tempted to give myself a downvote, totally missed phi was an actual constant in itself. http://en.wikipedia.org/wiki/Golden_ratio

Planck's constant is almost exactly 10^-33/(phi-0.108843677). Coincidence? I think not.