A brief summary of how accurately we have measured the fundamental physical constants that make up the laws of nature (i.e. the Standard Model, General Relativity, and a couple of constants from cosmology) is set forth below. It omits constants that have a value that is fixed theoretically to an exact number determinable without experiment in those models (e.g. the zero rest mass of a photon or the number of types of gluons or the number of generations of fermions or the charges and weak isospins of the fundamental particles). Experiments have essentially ruled out models that do not have exactly three generations of fermions and exactly three QCD colors.
Some of the greatest uncertainies regarding fundamental constants in the Standard Model involve neutrino physics because neutrinos are very difficult to measure directly.
Two key experimentally measured constants in the Standard Model: the hierarchy of the three neutrino mass eigenvalues ("ordinary" or "inverted") and the CP violating parameter of the PMNS matrix that governs neutrino oscillations are not known at all (although its other three parameters have been estimated to meaningful accuracy, as discused below).
There is a roughly 400% uncertainty regarding the absolute masses of the neutrinos (the sum of the three mass eigenvalues cannot be more than about 0.3 eV, but the difference in mass between the lightest mass eigenvalue and the heaviest puts a minimum value of the sum of the three masses at about 0.07 eV). This could be cut in half over the next several years to a decade by experiments currently under way.
PMNS matrix angle theta13 has an experimentally measured value with an uncertainty of about 13%. The mass difference between the second and third mass eigenvalues for neutrinos and theta23 have uncertainties of about 5%. Theta 12 and the mass difference between the first and second neutrino mass eigenvalues have uncertainties of about 3%.
Another uncertainties in neutrino physics in the Standard Model, are whether neutrinos are Dirac or Majorana in nature (if they are Majorana in nature, they give rise to two more unknown Standard Model constants for Majorana neutrino related CP violations).
Neutrinos, and in particular, neutrinoless double beta decay rates, are critical parts of models with lepton number violations and with Majorana neutrinos. This has not been discovered to date.
Substantial progress in pinning down all of these constants is anticipated over the next several years to a decade or so due to the concerted efforts of multiple active experiments investigating these matters.
A number of experimentally measured strong force physics, i.e. quantum chromodynamics (QCD) constants are also know with a fair amount of uncertainty. The masses of the quarks are known with the amounts of uncertainty shown parenthetically after each one (the maximal number if the uncertainty around the preferred value is asymmetrical): u (30%), d (15%), s (5%), c (2%), b (0.7%), t (0.5%).
Despite the uncertainty in the indirectly measured u and d quark masses (which are always "confined"), the mass of the proton and neutron made entirely of these quarks and gluons are known with great precision, as are the masses of most unstable mesons and hadrons made up only of up and down quarks.
The coupling constant is the strong force coupling constant known to about 0.6% accuracy.
In the big picture, this means that QCD calculations (i.e strong force physics) are accurate to predict many observables (e.g. proton mass) to only about a 1% precision, making first principles calculations far less accurate than direct experimental measurements in almost all cases.
There are some questions relating to the infrared behavior of the the strong force (i.e., is there a "non-trivial" infrared fixed point and if so, what is it)?
It is theoretically possible, without doing great injustice to the Standard Model, for there to be a CP violating constant in the strong force equations, although the current efforts to measure this are consistent with zero and the Standard Model assumes that it is zero.
Improvements in the measurement of these constants is likely to be incremental and more gradual than in neutrino physics, in part, because of the amount of progress that has been made to date. The measurements in these fields are more mature and much of the problem relates to developing accurate theoretical models of complex systems that are actually measured in order to make comparisons between theoretical predictions and experimentally measured values possible. Since quarks make up a quite small portion of the mass of mesons and hadrons (since most of the mass is attributable to the gluon activity that binds the quarks), theoretical predictions have to be very precise to translate these measurements into particle mass measurements.
Extensions of Koide's formula allow for estimates with precision on the order of the electroweak masses described below, but the lack of experimental precision makes these formulas difficult to verify experimentally. Also, some of these extensions seem to contradict the admittedly only marginally precise data that we have in hand to date for up and down quarks without further modification.
Electroweak constants (apart from neutrino physics)
The measured values of the CKM matrix mixing angles:
θ12 = 13.04±0.05°, θ13 = 0.201±0.011°, θ23 = 2.38±0.06°, and δ13 = 1.20±0.08.
The uncertainties, expressed on a percentage basis, in each respectively are to one significant digit: 0.4%, 5%, 3%, and 8%. The need to have consistency with an overall fit which is experimentally confirmed so far, also makes the whole somewhat less malliable than individual measurements of these constants by themselves, so the uncertainties of modestly overstated.
This Higgs boson mass is known to a certainty on the order of 2% and the vaccum expectation value of the Higgs field is known to a certainty on the order of 0.5%. The W boson mass is known to a precision of about 0.02%. The Z boson mass is known to a precision of about 0.002%.
The electron mass is known to a seven significant digit accuracy. The muon mass is known to a nine significant digit accuracy. The tau mass is known to a five significant digit accuracy.
The electromagnetic coupling constant is known to a ten significant digit accuracy. The weak force coupling constant is known to an eight significant digit accuracy.
The only electroweak physics left to refine are some modest improvements in the accuracy of the CKM mixing matrix angles that the LHC is facilitating as a side project, and increased confirmation of the Higgs boson mass and that its properties match those of the Standard Model Higgs boson (which so far have been satisfied in multiple respects to the limits of current experimental accuracy).
General relativity and cosmology constants
The speed of light in a vacuum is known to an accuracy of roughly nine significant digits. Planck's constant is known to an accuracy of about seven significant digits. The accuracy of Einstein's law of special relativity has been tested to accuracies comparable to those of the speed of light and the electroweak constants.
The gravitational constant is known to an accuracy of about one part per thousand. The cosmological constant in the equations of general relativity is known to only a roughly one order of magnitude precision: it is roughly 10−52 m−2. The Hubble constant and estimated age of the universe are known to a precision of about 1%.
Various deviations of general relativity from simple Newtonian gravity have been measured experimentally and found to match the predictions of General Relativity to precisions of one part per thousand to one part per ten thousand in most cases (e.g. the precession of the perihelion of Mercury's orbit, the deflection of light by the sun, the gravitational redshift of light, graviational lensing, light time travel delay, frame dragging, equivalence principal, binary pulsar behavior in strong fields including apparent gravitational waves).
Dark Matter and quantum gravity
Of course, the big unknown in astronomy, cosmology and gravitational physics is the question of dark matter about which there is no single consensus explanation for their mechanism. For example, there is no identified dark matter particle. The effects attributable to dark matter have been measured to precisions on the order of about 1%, but the details of their cause isn't known and more experiments seem to be creating more questions rather than more certainty.
Dark matter phenomena are definitely the greatest "known unknown" of physics. WMAP data fit dark matter cosmology models involving cold dark matter and a cosmological constant, and allow, but disfavor, up to one kind of light dark matter particle (although other sources of data pretty much definitively rule out another generation of neutrino as that particle).
Particle physics experiments and direct dark matter detection experiments have largely ruled out a large mass range of weakly interacting dark matter particle candidates.
The single constant in the empirical MOND model which quite accurately models essentially all galactric rotation curves observed by astronomers (regardless of the accuracy of the mechanism it implicitly assumes which is a modification to the law of gravity in weak fields rather than a dark matter particle) is known to an accuracy of about one signficiant digit. Different dark matter models have several constants each that are fitted to experimental data.
Searches for quantum gravity distinctions from general relativity has focused on black hole event horizons, implications for cosmological models, implications for the large scale structure of the universe, cosmic ray behavior, WMAP cosmic background radiation measurements, and weak field behaviors that could lead to phenomena equivalent to dark matter and dark energy (at least in part). Limited efforts have also been underway to measure gravitational effects at the sub-centimeter level which are difficult because gravity is so weak relative to other forces that could be relevant at such short distances. Some of the very subtle predictions of general relativity are very difficult to measure with human scale experimental resources. In understanding this difficulty consider, for example, that the entire planet Earth, for example, radiates about only about 200 watts of gravitational radiation
Theoretical considerations involved in harmonizing the Standard Model and General Relativity play a larger role than astronomoy observations in current efforts to develop theories of quantum gravity at the moment. The tricky part at this point is to devise ways to experimentally test possible variants on General Relativity.
Limitations on beyond the Standard Model physics
All experimental results to date are consistent with the Standard Model.
A few alternatives to the Standard Model are admitted by experiment, but are highly constrained experimentally at "low energies" (up to about 100 GeV-10 TeV), which a low relative to hypothetical maximal energy scales such a "grand unification" energies and the Planck scale, but exceed the energy scales seen in all but the most extreme processes (mostly not too long after the Big Bang) in nature.
At the currently established mass of the Higgs boson, the Standard Model is mathematically coherent up to roughly the Planck scale, something that would not have been the case at many other Higgs boson masses. So, beyond the Standard Model physics isn't strictly necessary for it to function even at very high energies, even if it is an "ugly" model that leaves the "why" questions behind many of its features unexplained.
Strict experimental bounds on lepton number violations, neutrinoless double beta decay, flavor changing neutral currents, proton decay, non-Standard Model particle decays, and minimum masses of hypothetical particles with particular properties greatly constrain these models, as does the discovery of the Higgs boson. Simpler supersymmetry models (including string theories), technicolor models, and extradimensional models, for example have well established experimental limitations placed upon them. Strict boundaries have been established on the constancy of a number of key fundamental physical laws and constants over billions and billions of years as well.
Any modification to the Standard Model must, essentially, add moving parts to a minimal Standard Model but retain much of its overall structure and manifest only in extreme conditions. Supersymmetry models, for example, have more particles than the Standard Model, but those particles all behave according to rules very similar to those governing the behavior of Standard Model particles.