There have been a bunch of questions related to the hierarchy problem, but I still can't help but feel that an assumption is being made that is not backed by any example of correctness and is thus potentially unwarranted. Of all of the questions already posted, I would say mine is most closely related to this one.
My understanding of the hierarchy problem is the following: in the context of the standard model thought of as an effective field theory with some cut-off $\Lambda$ (typically thought of as being far above the electroweak scale), the bare Higgs mass must be incredibly finely tuned so that it cancels its quantum corrections, quadratic in $\Lambda$, to leave behind the relatively small physical Higgs mass.
My issue with the supposed "problem" is this: why is the view that the standard model is an effective field theory with a cut-off any more valid than the view that it is a renormalisable theory where the cut-off is just a part of the regularisation/renormalisation scheme, eventually taken to be arbitrarily large (i.e. infinite)? In the latter picture, any contribution which is divergent with increasing $\Lambda$ is infinite as $\Lambda$ goes to infinity, so I don't see how a quadratic divergence is actually any worse than a logarithmic one.
The Higgs mass is the only parameter in the standard model that is dimensionful, so there is no other example to demonstrate that the idea that physical, dimensionful quantities should have values on the order of a suitable power of the cut-off is correct. In fact, outside of the standard model, the cosmological constant has the same problem; in these two cases of reasoning for the values of dimensionful parameters, the naturalness argument seems to fail terribly both times.