Is there any exact law in physics? Even for maybe the most fascinating computation in physics, about anomalous magnetic moment of the electron, theoretical physics gives:
$$a = 0.001 159 652 164 ± 0.000 000 000 108$$
which is not "exactly" true:
$$a_{exp} = 0.001 159 652 188 4 ± 0.000 000 000 004 3$$
so even laws of quantum field theory are not axact. (in second comment below Dvij D.C. said by this example there no violation and he is right: experimental interval is a subset of theoretical interval. but my question is still there because there are violation by other examples). In first page of "The Feynman Lectures in Physics" Feynman says: In fact,
everything we know is only some kind of approximation, because we know that
we do not know all the laws as yet. and in next page Finally, and most interesting, philosophically we are completely wrong with
the approximate law.
But what about some laws such as conservation laws of mass-energy, angular momentum, electric charge,... and sameness of gravitational and inertial mass and so on? is there any experiments which show even extremely small violation from this laws as we see in $a$-coefficient? if these are exact how do you interpret what Feynman says? could you give a list of exact laws in physic?
 A: There are two different things that you seem to be confusing. I will first discuss them and then come to the question you ask at the end.

A Theory That is An Approximation of Nature
This means that our theory only approximately resembles nature. The theory can still make exact predictions but those exact predictions might be slightly different from experimentally observed phenomena (depending on how well the theory approximates nature and how precise the experiment is). For example, Newtonian mechanics makes an exact prediction that the momentum of a particle with mass $m$ moving at a velocity $v$ will be $mv$. However, it only approximately describes nature.
There are a few things to notice here:

*

*There are such theories that we know are approximations of nature because we already have better theories to describe the same phenomena and we recover the former class of theories as an approximation of the latter class of theories. For example, we know that the Newtonian theory of gravity is an approximation because we can obtain it as the low-energy limit of the general theory of relativity which we know to be more exactly true than the Newtonian theory of gravity (e.g., it explains the precession of Mercury, correctly predicts the deviation of light around a planet, etc. -- things that the Newtonian theory of gravity fails to do).

*There are such theories that we expect to be approximations of nature even tho we don't have better theories to describe the phenomena that these theories describe but we have observed experimental deviation from the predictions of the said theories. For example, we know that the Standard Model of particle physics must be only approximately true of nature because while several incredibly precise experiments confirm its predictions, the observation of neutrino masses shows a deviation of nature from the predictions of the Standard Model (which predicts massless neutrinos).

*There are such theories that we expect to be approximations of nature even if we don't have either better theories to describe the phenomena that these theories describe or observed experimental deviations from the predictions of the said theories. This will be the case when we have theoretical reasons to expect that a deeper theory ought to exist of which the said theories will be an approximation. For example, (not all but most) physicists believe the general theory of relativity to be an approximate theory because it is a classical theory and we have good reasons to believe that a deeper quantum mechanical theory of gravity exists. We have neither detected an experimental violation of any prediction of the general theory of relativity (again, a minority of physicists believe that galactic rotation curves are such an observation but most physicists expect dark matter to explain them) nor do we have a deeper theory from which general relativity arises (of course, there are candidates for such a deeper theory, e.g., a string theory).

*Finally, if we ever arrive at a point in the history of science where we neither have a concrete theoretical reason to expect a deeper theory behind the best theory that we have nor do we have any experimental violation of any prediction of the theory, as good Baeysians, we should still not update our credence of our theory being an approximation to zero because that would mean that no future experimental or theoretical argument can convince us to modify our theory.  That is clearly a bad scientific position to hold. In other words, the best we can ever hope to say about a theory is that there is neither an experimental nor a theoretical reason to believe that the theory is not exact but we can't be entirely sure.

A Prediction That is An Approximation of the Theory
Regardless of whether a theory is exact to the best of our knowledge or it is known to be an approximation, the predictions that it makes can be either exact or approximate. This simply arises from our inability to solve the equations of motion of the theory exactly. For example, almost all of our predictions coming from the Standard Model are approximate because we either do the calculations perturbatively (i.e., very roughly speaking, we do a Taylor series expansion in a small parameter and truncate the series at some order of the small parameter) or because we do the calculations via numerical simulations.
My point here is that just because you see error bars on a theoretical prediction does not mean that the theory being used to make the prediction is not exact. It very well might be exact to the best of our knowledge, just that we don't know how to solve its equations exactly.

Finally, two points regarding the question you actually asked:

*

*As to whether there is any law that is exact in physics, all we can say is that there are certain things that we can expect to be exact with various levels of confidence but we obviously can't be sure (see the last bullet point in "A Theory That is An Approximation of Nature" section). For example, (local) Lorentz invariance is exact to the best of our knowledge. Now, in a quantum theory of gravity, the concept of spacetime might itself be replaced by something else and whether or not Lorentz invariance is maintained in some form is certainly beyond my paygrade to predict. Another aspect of our theories that we expect to be exact is the unitarity of quantum mechanics -- however, it might as well be subtly modified in a full quantum theory of gravity, I don't know.


*As to whether there is any prediction that is exact within our current theories, absolutely! The ratio of inertial to gravitational mass is exactly one in general relativity which is our best theory of gravity. The conservation of electric charge is exact in the Standard Model because the global $U(1)$ is an exact symmetry in the Standard Model, etc.
A: A piece of the context:

You might ask why we cannot teach physics by just giving the basic laws on page one and then showing how they work in all possible circumstances, as we do in Euclidean geometry, where we state the axioms and then make all sorts of deductions. (So, not satisfied to learn physics in four years, you want to learn it in four minutes?) We cannot do it in this way for two reasons. First, we do not yet know all the basic laws: there is an expanding frontier of ignorance. Second, the correct statement of the laws of physics involves some very unfamiliar ideas which require advanced mathematics for their description. Therefore, one needs a considerable amount of preparatory training even to learn what the words mean. No, it is not possible to do it that way. We can only do it piece by piece.
Each piece, or part, of the whole of nature is always merely an approximation to the complete truth, or the complete truth so far as we know it. In fact, everything we know is only some kind of approximation, because we know that we do not know all the laws as yet. Therefore, things must be learned only to be unlearned again or, more likely, to be corrected.

So we know that what we don't know all the laws yet. And therefore, everything we know is an approximation only. There is an expanding frontier of ignorance, and I think he means the frontier that exists when looking to small scales. Of course every expanding sphere of knowledge brings a frontier along. Separating what's outside that sphere and inside of it. There are more things outside of that domain than inside, and maybe each new piece of the domain of nature has its own laws.
We can't know everything, but it's imaginable that when we look to smaller and smaller scales, some ultimate truth can be found (Popper would call such a theory non-scientific, but how can you falsify something if you have hit rock bottom so to speak?).
You can't, obviously, just state the laws of nature. You have to grow into these laws, and once you're in you have to realize that what you've learned is an approximation only. Again, I'm not sure if he means the laws at the bottom or higher level laws, which are approximations always. They are still laws and they can operate quite independently of the laws deep down.
The laws on the smallest scale, which are what we call fundamental laws, can't be exactly right yet and we know it, according to Feynman. So we must let go what we have learned, or at least correct it and see it as an approximation to the deeper theory.
I'm not sure if he thinks that a final theory is possible. I don't see why not, and when we think we have found one, it will always be nature that makes the final call, like in all physical laws we create or find out about.
A: Any theory or law in physics is only as good as its experimental validation. Every experimental measurement has a non-zero experimental error. Therefore no theory or law in physics can be validated exactly.
The speed of light in a vacuum may vary in its $20$th digit. The law of conservation of energy may only be correct to one part in $10^{50}$. The second law of thermodynamics may be violated one time in every $10^{100}$.
All physics can aspire to is to set down laws and principles that are correct up to the limits of experimental error - and to devise ever more ingenious experiments to reduce the size of this experimental error.
