The Standard Model has many parameters that must be determined experimentally, although there is not a unique way to describe these constants as some constants are functions of other constants. For example, one could choose to make either the Higgs vev or the weak force dimensionless coupling constant part of your fundamental list of constants and derived the others from that. Similarly, you define either the dimensionless Higgs boson coupling constants called Yukawas, or the pole masses of the fundamental fermions and bosons, as the more fundamental quantity.
One way to summarize the list of experimentally measured constants in the Standard Model is that there are 12 fundamental fermion masses, 3 fundamental boson masses, 4 CKM matrix parameters (which govern the probability that quarks transform into different kinds of quarks), 4 PMNS matrix parameters (that govern the probability that neutrinos oscillate to different neutrino masses), and three standard model force coupling constants (for the electromagnetic, weak force and strong force), for a total of 23 experimentally measured parameters, plus some additional more general physical constants determined experimentally, including, at a minimum, Planck's constant and the speed of light in a vacuum, for a total of 25. This list, however, is somewhat redundant. One of the fundamental boson masses (either the W or the Z boson) can be derived from the other of the electroweak boson masses and the electromagnetic and weak force coupling constants, bringing the list to 24.
Some of the most solid relationships conjectured below can eliminate one parameter each from the CKM matrix and PMNS matrix (the theta13 angle in each of them), the tau lepton mass (predicted by Koide's rule to be 1776.96894(7) MeV, which is consistent with the direct experimental measurement which has an uncertainty of 0.12 MeV, but much, much more precise), and the top quark mass (predicted to be 173,666(125) MeV based upon the relationship of the sum of the square of the fundamental particle masses given their measured values to the square of the Higgs vev, consistent with the direct measurement at 1.67 sigma, but with a precision better than the plus or minus 400 MeV in the directly measured experimental value) from the list, in theory reducing the number of truly independent experimentally measured parameters to 20. Less solid relationships, if developed further and validated, could further reduce this list substantially.
Gravity as explained through general relativity, has two more experimentally measured constants: the gravitational coupling constant a.k.a. Newton's constant "G", although it can be defined in one of several dimensionless ways, and the cosmological constant a.k.a. lambda (which is the most common, but also disputed explanation for phenomena attributed to "dark energy" that are not well understood at a fundamental level). This would bring the total number of fundamental constants of physics to 27.
Some dark matter and/or modified gravity theories that seek to explain dark matter and/or dark energy phenomena, dispense with the cosmological constant and/or add one or more experimentally measured fundamental constants to explain dark matter and/or dark energy phenomena (and sometimes one more additional fermion and/or boson types as well).
Why these parameters take the values that they do is an open question that is mostly unanswerable at this time. But, almost everybody involved in this branch of physics intuitively and personally believes that these relationship are not just random and have some cause based upon some deeper theory that is not currently known to us.
* The CP violation parameters in the CKM matrix and PMNS matrix are an effect that is actually independent of the other three parameters in each matrix. It also is worth noting that the CP violation phase could be the same in the CKM matrix and PMNS matrix given the great uncertainty involved in the PMNS matrix value. In the CKM matrix, the CP violating phase is δ13 = 1.20 ± 0.08 radians (i.e. 68.8º plus or minus 4.6º). In the PMNS matrix, the mean measurement for the CP violating phase is 246º subject to very large error bars that make the central number measurement not very meaningful. The two values could also be complementary and actually add up to 360º.
One possibility that seems plausible is that the CP violation in both cases arises through W boson interactions (and possible a parallel fundamental boson currently unknown related to neutrino oscillation), which take place only with left parity matter and not with right parity matter. It further seems plausible to me that CP violation is maximal at tree level in the relevant interactions, and that deviations from maximal CP violation in W boson interactions arise from higher order loops.
* I strongly suspect for symmetry reasons and because massless particles do not experience the passage of time in the way that massive particles do, that any force mediated by a massless boson (i.e. electromagnetism mediated by photons, the strong force mediated by gluons with zero rest mass, and quantum gravity mediated by massive gravitons) are necessarily not CP violating as a result.
Similarly, particles that do not have electromagnetic charge and also decay in a matter-antimatter preserving fashion, like the massive Z boson and the Higgs boson, can likewise not violate CP on symmetry grounds (although Z bosons only interact with left parity particles which may be more subtly CP violating in a sense).
* I suspect that all dark matter and dark energy phenomena will ultimately be explained as quantum gravity effects, and that dark matter particles (other than massless gravitons) will be ruled out. I suspect that the only BSM particles, other than massless gravitons, that have not yet been discovered, are limited to a possible neutrino oscillation boson, possible right handed neutrinos and left handed antineutrinos with the same mass as their parity partners and no Standard Model interactions, and some set of particles (less numerous than the current set of Standard Model particles) that give rise to the Standard Model particles (with the caveats noted above, if necessary) and no other particles whatsoever, such as fundamental string-theory like strings or some kind of preon.
* I suspect that baryon number and lepton number are conserved in all processes except sphaleron processes (if those actually even exist, and in which case B-L is still conserved), and possibly in the graviton equivalent to photo-production of particles. Hence, neutrinoless double beta decay does not occur, and neither do proton decay, or at the tree level, flavor changing neutral current processes.
* The conventional explanation of neutrino oscillation which is sufficient as a phenomenological theory is that there is a mismatch between three electroweak neutrino flavors and three mass eigenstates causing them to oscillate. But, the possibility that neutrino oscillation is actually mediated by a new fundamental boson analogous to the W boson, or that it involves a combined virtual W+ and W- boson loop, are possibilities that have not been ruled out to my satisfaction so far that could provide a first principles explanation of some of the associated physical constants that are measured experimentally.
* I seriously doubt that neutrinos are Majorana particles that are their own anti-particles, even though they lack electromagnetic charge and appear to come only in left handed neutrinos and right handed anti-neutrinos which would be inconsistent with the conventional Higgs mass generation mechanism. I likewise find a seesaw mechanism to be highly implausible. I do think it is plausible that there exist right handed neutrinos with the same mass as their left handed counterparts that don't interact with any of the three Standard Model forces and instead merely interact with their left handed counterparts, and left handed antineutrinos that are analogous, that make it possible for neutrinos to have a standard Higgs mechanism Dirac mass just like all of the other fundamental fermions in the Standard Model. I strongly suspect that there are no "sterile neutrinos" and that the "reactor anomaly" that suggested that they might exist with a mass on the order of 1 eV is actually just a fluke or an experimental design problem.
* In both the CKM matrix and the PMNS matrix, ignoring CP violation, the probability of a first to second generation transition times the probability of a second to third generation transition, is equal to the probability of a first to third generation transition. Thus, these are actually matrixes with at most three independent parameters (including CP violation), not four. It is possible that these two remaining generation transition parameters can even be reduced to two experimentally measured parameters or even just one for both matrixes, but simply reducing them from three to two for each matrix would represent scientific progress. If the same one or two parameters can be used to explain generation transitions in bot the CKM matrix and the PMNS matrix, this would probably be due to a concept known as "quark-lepton complementarity" since the sum of the CKM and PMNS matrix mixing angles for theta12 are fairly close to a combined 45º (which is the maximal mixing angle), as are the the sum of the CKM and PMNS matrix mixing angles for theta23.
In the case of the PMNS matrix, applying this formula with angles in radians, this implies a mixing angle from the first to third generation of 7.994º, while the measured value is 8.54º plus or minus 0.15º. Given the uncertainties in theta12 which is 33.62º plus or minus about 0.77º and theta 23 which is 42.8º plus or minus about +1.9º/-2.9º, this results are consistent at two sigma.
In the case of the CKM matrix, this implies a first to third generation mixing angle of 0.172º compared to the measured value of 0.201º plus or minus 0.011º which is also consistent at two sigma due to the uncertainty in this value combined with the uncertainty in the other two measured values that enter into the calculation.
Doing a global fit of the CKM and PMNS parameters with these constraints would be interesting and informative.
This relationship also suggests that the probability of a fermion generation change in logically prior in a deeper theory to the masses of the particles at particular generations.
* It is plausible to me that there may be some functional relationship between the Cabibbo Angle (of the CKM matrix a.k.a. lambda in the Wolfenstein parameterization) and Weinberg Angle (which pertains to the relative masses of the W boson and the Z boson as a result of fundamental relationships between electroweak theory quantities), as they are numerically quite similar and both involve the weak force, although they are not similar enough to each other to have values consistent with each other given current measurement precision.
This observation and the fact that all fundamental particles that interact via the weak force have a rest mass, while all fundamental particles that do not have rest mass do not interact via the weak force, makes wonder if the W boson plays a more central role in generating the fundamental particle rest masses (including the neutrino masses) and the Higgs boson is less important in this process than commonly assumed, with the W boson dynamically balancing the fundamental fermion masses.
* The overall mass scale of the Standard Model fermions is probably a function of the Higgs vacuum expectation value (which is itself intimately related at a functional level to the weak force coupling constant) since the sum of the square of the fundamental fermion pole masses is equal to the square of the Higgs vev consistently up to a 1.3 error in the measurements, with the uncertainties in the top quark mass measurement (85.2% of the uncertainty) and the Higgs boson mass measurement (13.6% of the uncertainty) dominating that uncertainty, and uncertainty in bottom quark mass (0.6% of the uncertainty), the charm quark mass (0.5% of the uncertainty) and the W boson mass (0.1% of the uncertainty) accounting for almost of of the remaining uncertainty in the comparison of the experimentally valued masses to the experimentally determined value of the Higgs vev (which is known to a precision of one part per 246 million).
This also explains why the Higgs boson has the mass that it does (to fill the gap not filled by the other fundamental particles of the Standard Model). From this perspective, the Higgs boson mass is very "natural". The "hierarchy problem" related to the Higgs boson mass is simply a function of an unnatural way to think about the means by which the Higgs boson mass arises.
If this is the explanation of the overall mass scale of the rest masses of the Standard Model fundamental particles, it appears that the fundamental fermion masses account for slightly less than half of the total, while the fundamental boson masses (which are known more precisely) account for slightly more than half of the total (something that wouldn't change if there was a massless graviton).
The best available estimates of the top quark mass are a bit too low (by 2.6 sigma) and the best available estimates of the Higgs boson mass are a bit too high (by 3.3 sigma), both of which would be necessary to make them exactly equal, and since both errors are independent and both have to be correct, the combined deviation of that theoretical prediction is very significant (between 4 sigma and 5 sigma). This almost symmetry between fermions and bosons (in which the squared masses of the bosons accounts for about 51% of the total and the squared masses of the fermions accounts for about 49% of the total), which might be exactly equal at some energy scale greater than pole masses, perhaps the Higg vev energy scale, since boson masses for the most part decline with energy scale faster than fermion masses do, may explain why supersymmetry provides the insights that it does and also why supersymmetry itself is not necessary.
Even if the sum of the square of the fermion masses turn out to not be exactly equal to the sum of the square of the fundamental boson masses, the more general relationship between fundamental fermion pole masses and the Higgs vev that is empirically well established to the limits of current measurement precision, would at least explain the magnitude of the top quark mass as a "filler" after all of the other fundamental fermion masses are accounted for.
This analysis, if theoretically sound, is also one of the more fruitful arguments to rule out the existence of new heavy fundamental particles types, particularly as top quark, Higgs boson and W boson mass measurements grow more precise. If the Higgs vev really is equally to the square of the fundamental particle masses, the uncertainties in the known fundamental particle masses leave no room for particles with masses over a few GeV that have mostly been ruled out in direct searches, to have been omitted.
* The rank order of the mixing angles of theta12 and theta23 in the CKM and PMNS matrixes has a relationship to the magnitude of the mass ratios of the starting and ending points involved in those transitions (adjusting in some appropriate matter, such as a geometric mean, for the fact that the CKM matrix involves two sets of mass differences per generation and not just one). Likewise, by some appropriate measure, bigger differences in mixing angles correspond to bigger differences in mass ratios of the starting and ending points involved in those transitions. Note that this description very carefully avoids stating a particular functional relationship, which is unknown.
This implies that the ratio of third generation to second generation quark partners in W boson transitions is higher than the ratio of second generation to first generation quark partners in W boson transitions since the mixing angles are 2.38% and 13.04% respectively. The respective geometric means in those case are about 275 and 100 respectively, and the ratio of the mixing angles is about 5.5.
The neutrino mixing angle for the third generation to the second is about 42.8% (assuming a first quadrant value), and the neutrino mixing angle for the second generation to the first is about 33.62% , and the ratio of the mixing angles is about 0.79. So, the ratio of the third generation neutrino mass to the second generation neutrino mass (which is about 5.6 or less), should be smaller than the ratio of the second generation neutrino mass to the first generation neutrino mass.
If the pattern of the CKM matrix were to hold, one would expect the ratio of the second heaviest neutrino mass to the lightest neutrino mass to be about 9.
Assuming the increasingly experimentally favored normal hierarchy, and given that the mass difference between the heaviest neutrino mass and the middle neutrino mass is about 49.4 meV and the difference between the middle neutrino mass and the lightest neutrino mass is about 8.7 meV, one would expect the lightest neutrino mass to be a little less than 1 meV, but probably not much lower than 0.5 meV.
This implies absolute neutrino masses of about 0.9 meV, 9.6 meV, and 59 meV, with a sum of the three neutrino masses equal to about 69.5 meV plus or minus about half an meV. This is very close to the minimum mass possible for the sum of the three neutrino mass eigenstates, given what we know already.
* The difference between the electron mass and the lowest neutrino mass is due fundamentally in some manner (possibly due to their respective self-couplings) to the ratio of the electromagnetic force coupling constant to the weak force coupling constant which is of approximately the same order of magnitude. Likewise the electron and up quark masses may be a function of their self-couplings, although the mechanism by which higher generation fermions acquire their masses and why they have only three generations, is still somewhat mysterious even if we can come up for a mathematical formula that accurately determines the fundamental fermion masses.
* Lepton universality is probably only an approximate rather than an absolute symmetry that holds only because the ratio of the mass of each charged lepton to the mass of each charged lepton less the corresponding neutrino mass is so close to 1 for all three of the charged leptons. Violations of lepton universality in excess of this magnitude is probably due to experimental and/or theoretical error.
* The fact that Koide's Rule holds for the charged leptons to the limits of experimental accuracy, is probably for fundamental reasons similar to those for lepton universality and the fact that there are only three charged leptons.
* The ratio of the quark masses appears to be, at first order, the product of something very close to Koide's rule, but adjusting for the possibility that there could be transitions other than the most common one implicated by Koide's rule implemented directly, by making an adjustment of an order of magnitude equal to the mass difference of the omitted transition time the probability of the omitted transition taking place under Koide's rule.
* I think that it is plausible that the Standard Model formulation of the fundamental equations and axioms of quantum chromodynamics a.k.a. QCD a.k.a. the modern explanation of the strong force that holds protons, neutrons and other hadrons together, is incomplete and missing a rule or two, or a key axiom or two. For example, I would not be surprised if a missing axiom established that free standing glueballs were impossible for some reason.