Why do the masses of fundamental particles seem to increase exponentially? The (15 positive) masses of fundamental particles are measured inputs to the standard model. They seem to increase exponentially when ranked in increasing order, or perhaps follow a power law when ranked in decreasing order. The masses (in GeV/c^2) are approximately:
0, 0, <.000001, <.00017, .000511, .0022, .0047, <.0182, .096, .10566,
1.28, 1.7768, 4.18, 80.39, 91.19, 124.97, 173.1
where the inequalities are upper bounds for the unknown neutrino masses. I don't want to stray too far into numerology, but when the logarithm is plotted (excluding the photon and gluon) you get a pretty straight line (-14.3788+1.24443n):

I'm aware of the Koide formula, but haven't heard of this little phenomenon before. Does the Standard Model have anything to say about this, perhaps bounds on the masses, or are these really just coincidences that will only be explained when there is a deeper theory?
EDIT: To address the comments, I've removed the two gaps I put in by hand where there are subjective jumps between particles 10 and 11, and particles 13 and 14. The masses are in rank order, which is unique up to ascending or descending. I have also included the norm of residuals as the 'error'. To be clear, this is an approximation. The Kolmogorov complexity of the complete list (upper bound computed using Mathematica, $ByteCount[Compress[list]]$) is approximately 256, with an error of 0 (really a small positive $\epsilon$) using that description. With this description, the K-complexity is approximately 112, and the error is 4.31219. The comments seemed to imply that if you allow gaps, one could bring the error to zero. However, there is a cost in K-complexity. Here is a plot:

EDIT 2: To address a comment, I have colored the upper bounds for the neutrino masses to clarify that they are unknown. This increases the K-complexity of the description from 104 to 144.
 A: First off, the right part of the graph is not really contributing anything: the $W$-boson, $Z$-boson, and Higgs boson all have mass in the same range, because these masses all come from the scale of electroweak symmetry breaking. Similarly, the left part isn't saying much, because everything is zero or too low to measure. So we're left with the $6$ quark  and $3$ charged lepton flavors.
Explaining their masses is known as the flavor problem. Unfortunately, even though it really seems like there's something going on here, nobody has succeeded in writing down a compelling theory. Literally thousands of attempts have been made, but the resulting models generally are quite contrived, and either don't make any sharp and reasonably testable predictions, or make incorrect predictions. So most of these models don't attract much attention, for the same reason the Koide formula doesn't -- they're interesting, but it's not clear what more we can do with them.
A Taste of Flavor Model Building
People can spend decades working on this problem, and there's a great diversity of approaches, so I'll just give a taste. For a brief review, see here.
In the Standard Model, the quarks and leptons get mass through the Higgs mechanism. The symmetries of the Standard Model allow these particles to interact with the Higgs field, by emitting or absorbing one Higgs boson. When the Higgs field gets a vacuum expectation value, this turns into a particle mass.
The popular Froggatt-Nielsen mechanism is a twist on this. A new Higgs-like field $S$ is added and a new symmetry is postulated, so that different particles can interact with the $S$ field, but only by emitting or absorbing $n$ $S$ particles at once, where $n$ depends on the particle. The resulting mass scales as $\epsilon^n$ for some small parameter $\epsilon$, and fixing the $n$'s by defining the symmetry just right, you can reproduce the exponential structure in the quark masses. The quantitative agreement ends up about as good as your line. But then one could complain that this just reduces the question to why the symmetry has to be that way.
Another idea along these lines is radiative mass generation, where again the symmetries are fixed to give a hierarchical structure. Here, the heaviest generation (top, bottom, tau) can get masses at leading order in perturbation theory ("tree level"), but the second generation can only get masses at next-to-leading order ("one loop") and the first at the next order after that ("two loop"). Again, you need to set up the symmetries in a somewhat Rube Goldberg-esque way to make this happen.
In general, if you want to solve the flavor problem, then grand unification is useful, because it gives you a relation between quark and lepton masses in the same family. For example, in the simplest possible $SU(5)$ GUT, we have a relation between the positive and negatively charged quark masses in each generation,
$$\frac{m_b}{m_\tau} \approx \frac{m_s}{m_\mu} \approx \frac{m_d}{m_e} \approx 3.$$
This relation is not very accurate, but you can see how the rest of the reasoning would go: add some more Higgs fields to fix up the mass relations within each family, then combine it with one of the previous ideas to get an exponential hierarchy between families, and we've explained the pattern! The only problem is that in the process, we've introduced far more parameters than we managed to explain. Moreover, grand unification essentially always comes bundled with weak-scale supersymmetry, so many of these models have been rendered irrelevant by the LHC.
A completely different route is to appeal to the anthropic principle, or to cosmology. You can argue, for example, that the up and down quark masses can't be too far apart, or else you wouldn't get the rich structure of nuclear physics. Similarly, you can't adjust the electron mass too much without messing up the structure of chemistry. And the top quark, since it couples by far the strongest to the Higgs, determines the stability of the vacuum of our universe. But I'm not aware of any way to use these ideas to solve the whole flavor puzzle, because the other $8$ masses have essentially no impact on everyday life or cosmology.
The flavor problem is infamously hard. At the end of the day, we don't have anything that's essentially better, scientifically, than the line you drew. But lots of people are working on it!
A: The problem of this presentation is that it uses particles not showing the entry (neutrinos) as upper bounds, and for quarks, these are values at an arbitrarily chosen scale of $2~\rm GeV$: - the masses at $1~\rm GeV$ or $3~\rm GeV$ would also be interesting so really for up-quark=$.0022\pm 100\%$, down-quark=$.0047\pm 90\%$, strange quark= $.10566\pm50\%$,  Charm-quark=$1.28\pm30\%$.  So a plot as presented done correctly can be fitted by any monotonic function (monotonic since ordering is with mass value, thus arbitrary). Oh, I forgot to mention that bottom quark has ist problems to, any value between 3.5 and 5.5 GeV will make everyone happy..... So all we learn here is that if one orders numbers in sequence of value the resulting function is monotonic. And if I can be of help to anyone looking at W/Z (80/90 GeV, take geometric mean), Higgs (124) and top (174) please remember that there is a surprize: these are indeed in geometric progression with $\sqrt{2}$ factor and on top of evrything the minimal coupling of top is 1. So we know the Standard Model has a secret for us, many are looking to solve the riddle and travel to some nordic country in December.
A: For what it is worth. If you take the two different kinds of quarks from all the families and plot the log of their masses as a function of family number then you get approximate straight lines. For the charged leptons you can also get a straight line in this way if you assume four families with the second one missing.
Another way to look at this is to compute the following ratios for two types of quarks:
$$ N_u = \frac{\log(m_c)-\log(m_u)}{\log(m_t)-\log(m_c)} \approx 1 $$
$$ N_d = \frac{\log(m_s)-\log(m_d)}{\log(m_b)-\log(m_s)} \approx 1 $$
And for the leptons:
$$ N_e = \frac{\log(m_{\mu})-\log(m_e)}{\log(m_{\tau})-\log(m_{\mu})} \approx 2 $$
What this means, or whether this means anything, I don't know. I don't think there are any people that are trying to figure it out either.
