Why do we have a TeV scale? When model building we don't want to introduce any new scales into our theory. We usually try to have new particles at the Higgs (TeV) scale (to solve the hierarchy problem), at the GUT scale, or at the Planck scale. 
However, if the Higgs VEV already gives us a new scale, why would there not be new particles at some intermediate scale, for example say at $10^{5}$ TeV? In other words, what is unnatural about adding in new scales beyond the 3 that we are used to?
 A: Actually, the Higgs scale is not the TeV scale. The Higgs scale is the scale of electroweak symmetry breaking, i.e. $\mathcal O(100 \mathrm{GeV})$.
The Terascale comes into play along with the Higgs, as supersymetry -  the most popular extensions of the Standard Model - would actually like a small Higgs mass, much smaller than its measured value ($< M_Z$ to be more precise). In order to have a Higgs mass at $125$ GeV, we need supersymmetry breaking parameters to be on the order of TeV at least (disregarding extremely specific scenarios). Higher scale supersymmetry is still possible, but would not be as attractive from a more fundamental point of view.
There are also other effects, like the running of the strong coupling constant that could also hint at new physics, if we found a deviation at higher energies. Or gain further information on the viability of ideas such as Grand Unification.
Most importantly, with the LHC we can actually measure at the TeV scale, i.e. models and ideas taking place there have a realistic chance of getting verified or falsified in the forseeable future. I think that's the main reason, why Terascale physics is so important right now.
Edit - for more information: Fundamentally, there is nothing wrong with having many scales. Still, it would be hard to justify a world with effects at many different fundamental scales. The scale at $E \sim 0$ comes free, as this is the only energy that is fundamentally different from all others. This scale includes electromagnetic and strong force effects, i.e. all atomic and most nuclear dynamics (as they arise from interactions with massless force carriers). Then, we have one other scale for free, just because we see that there IS mass in the world. But looking at particle physics and gravity, we already have two: The electroweak scale $E \sim 100$ GeV and the Planck scale $E \sim 10^{19}$GeV. Quantum corrections want to equalize the scales, unless there is a symmetry protecting the quantity in question. That's one of the reasons supersymmetry is so popular: It adds a symmetry to protect the Higgs mass (which should be of the order of the Planck scale otherwise). Often, we theorists add the GUT scale $E \sim 10^{15}$ GeV to the picture as an intermediate scale, since the three forces of the Standard Model are (approximately) equally strong there. Then we already have four scales (of which we need to explain two). Now add the Terascale for supersymmetry breaking and we have five (three to explain). This is unsatisfactory, but alas, it's the best we can do for now.
A: If you want the new physics to solve the hierarchy problem, it's best if it is close to the weak scale, or else you will be left with a residual little hierarchy.
You are describing the "big desert" between the weak and GUT scales. I think it was motivated by the idea that SUSY lived at the weak scale, solving the hierarchy problem and insuring gauge coupling unification. Any physics between those scales would be unnecessary, could spoil unification or induce FCNC and proton decay.
I don't think the big desert hypothesis is particularly compelling, especially in light of the LHC results, but it is plausible.
