Naturalness, simplicity and SUSY With the aid of current LHC + (all the known HEP experiments)+ (astrophysical and additional laboratory measurements)...
Is SUSY natural? Is it simple? More specifically, what reasons people had (or have) in order to argue the "naturalness" of SUSY? Why (despite the fact it doubles the particle spectrum!) some people says that SUSY is "simple" or a simple solution to the "most" important theoretical issues in physics?
Bonus question: can a non minimal supersymmetric model be both "natural and simple", according to these definitions?
I define naturalness in the sense of 't Hooft: a QFT is natural when its coupling constant is of order one (with respect to, e.g., the electroweak -EW- scale). 
A theory is "simple" when it reduces the number of degrees of freedom of previous "effective" theories. 
In my (completely subjective) opinion, current data strongly suggest that:
i) SUSY is not natural. There is a relatively "large" gap between the electroweak scale and the SUSY breaking scale, if it (SUSY) exists...Then, the coupling constant of SUSY is not of order "one" with respect to the EW scale. The size of the SUSY breaking scale makes it to suppress the production of SUSY particles. 
ii) SUSY is not simple. People use to argue...And I agree with it...that SUSY provides a natural lightest supersymmetric particle (LSP), and thus there is a natural Dark Matter candidate. It also "unifies" forces in such a way the 3 SM forces and their coupling constants unify at certain large energy. The unification of coupling is "simple" and usually it is given as  a "proof" that SUSY is OK. Maybe it is true. However, gravity is not included in that calculation. I mean, I have not seen a calculation of the running of G in the framework of SUSY. Moreover, despite its goodness providing a natural particle/field for the dark matter, SUSY enlarges the number of particles (and the degrees of freedom) without (yet, please, I don't give up the idea of SUSY myself, I am trying to offer a cold criticism of its problems) experimental evidence beyond that dark matter we do not know what it is made of... SUSY is NOT simple because, from the purely mathematical viewpoint, it enlarges the number of free parameters and particles with respect to the SM. 
 A: 
A theory is "simple" when it reduces the number of degrees of freedom of previous "effective" theories. 

I do not really think that this statement  categorizes simple theories: the degrees of freedom of "effective theories".
Simplicity for a physicist is much more connected to an esthetic attribute than the degrees of freedom.
The heliocentric system is simple in the newtonian formulation because we find it esthetically satisfying. The exact same number of parameters and degrees of freedom have to enter for a  geocentric system, but certainly if physicist voted for simplicity they would go 100% for the heliocentric one.
Quantum mechanics is not a simple theory as far as a definition by degrees of freedom go. It has introduced the uncertainty principle multiplying the necessary degrees of freedom for a particle. Nevertheless nuclear and  particle physicists find it simple and beautiful in its description of nature. Simplicity is  like beauty in the eye of the beholder. 
When I was a graduate student back in the '60s we got a field theory for nuclear physics with super symmetry  for the nuclear levels. The professor found that it simplified certain branching rations and behaviors simply.
Thus in my opinion, if super symmetry is discovered at the LHC during the next 10 or 15 years , we will all find it simple, for eliminating hierarchy problems and infinities at high energies and offering predictions for further studies ( and maybe establishing strings: what can be conceptually simpler than a vibrating string?).
If you look at the isotopic table of elements there is little simplicity there, nevertheless complicated theories bring order out of the confusion.
The standard model rests on less than thirty  elementary particles, 3 neutrinos, 3 leptons,8 gluons, nine colored quarks , a photon, a Z and three Ws. So what if they are doubled? They are still less than the isotopic table of elements in complexity and ruled by simple rules. Who cares for the multiplicity of resonances etc? After all there are about 10^23molecules in a mole. Does this mean that the theory that describes their behavior is not simple?
Now natural is another story, and it depends on the surprises of nature. Is it natural that humans exist starting from a single complicated DNA molecule?
Natural is as natural it appears.
A: I would like to clear up the idea of "naturalness." I disagree with the current answer - it is not an aesthetic criterion. "Naturalness" and "simplicity" can be precisely and numerically defined with "Bayesian evidence", defined schematically
$$
\mathcal Z = \rm p(Data\,|\,Model)\\ 
           = \int p(Data\,|\,\textrm{Model's parameters and Model})\times p(\textrm{Model's parameters})\,d(\textrm{Model's parameters}).
$$
Within Bayesian statistics, models with greater evidence are more probable than those with small evidence.
A "natural" model agrees with experimental data across its    parameter space - it does not require its parameters to be fine-tuned    for its predictions to agree with experimental data.
A "simple" model makes robust predictions for observables that do    not change wildly within its parameter space (but don't necessarily agree with experiment).
Both of these notions are measured numerically by the Bayesian evidence. As an example, consider plotting evidence as a function of data for three models, as below. Note that the evidence is a pdf as a function of the data, so models have a finite probability mass to spend.

The observed data is marked on the $x$-axis. The model marked $H_1$ is the simplest, because it concentrates its probability mass about a precise prediction. But it's unnatural, because its evidence at the observed data is tiny. Carrying on with this reasoning, we see that $H_2$ is the second simplest model and the most natural model.
