Has unnaturalness motivated new ideas in physics?

The cosmological constant problem arises because the natural scale of the cosmological constant is $10^{120}$ times larger than what we observe. This implies that the dimensionless constant is much smaller than one and that is considered to be unnatural (though that's a perfect logical possibility). The hierarchy problem is of a similar nature. My question is whether a similar kind of reasoning motivated by unnaturalness has been useful to physics in the past. I.e., is there any example where some constant was deemed unnatural and to resolve the issue new physics was discovered? I would like examples from settled physics rather than from speculative theories. (So supersymmetry doesn't count.)

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I don't think there are good examples of "unnatural physics" in the past, because of all established physics, except for the CC and EWSB, is "natural". –  Vibert Mar 25 '13 at 7:41
That's what I suspected. I don't know how much faith to put into this sort of naturalness arguments as they are more philosophical than physical. I was trying to find out if there have been any precedents. From the long silence to my answer, apparently there is not. –  Sudip Paul Mar 28 '13 at 7:23