Why do we have an elementary charge but no elementary mass? Why do we have an elementary charge $e$ in physics but no elementary mass? Is an elementary mass ruled out by experiment or is an elementary mass forbidden by some theoretical reason? 
 A: Mass can't be quantized because the contribution of a particle to a system's mass is not a scalar, but a 0 component of a 4-vector, so if you have a system of quantized mass particles, their bound states would not obey mass quantization.
In semi-classical gravity, there is a simple reason that charge has to be quantized. If the proton had a charge infinitesimally bigger than the positron, you could make a black hole, throw in some protons, wait for an equal number of positrons to come out in the Hawking radiation, and then let the resulting wee-charged black hole decay while throwing back all the charged stuff that comes out. This would produce a small mass black hole with charge equal to any multiple of the difference, and it could not decay except by undoing the process of formation. This is obviously absurd, so the charge is either quantized or there are particles of arbitrarily small charge.
Further, the small charge particles can't be too heavy, since the polarizing field of the black hole with these wee-charges must be strong enough to polarize the horizon to emit them. If their mass is bigger than their charge, then they are net-attracted to the black hole, which causes a constipation for the black hole--- it can't get rid of its charge. So the wee charged particles must be lighter than their mass generically.
These types of arguments reproduce the simpler swampland constraints. That out universe is not in the swampland is the only real testable prediction that string theory has made so far (for example, it excludes models where the proton stability is guaranteed by a new unbroken gauge charge).
A: Let me add two references to points already mentioned in this discussion:
Today, there is no reason known why the electric charge has to be quantized. It is true that the quantization follows from the existence of magnetic monopoles and the consistency of the quantized electromagnetic field, which was shown first by Dirac, you'll find a very nice exposition of this in


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*Gregory L. Naber: "Topology, geometry and gauge fields." (2 books, of the top off my head I don't know if the relevant part is in the first or the second one).


AFAIK there is no reason to believe that magnetic monopoles do exist, there is no experimental evidence and there is no compelling theoretical argument using a well established framework like QFT. There are of course more speculative ideas (Lubos mentioned those).
AFAIK there is no reason why mass should or should not be quantized (in QFT models this is an assumption/axiom that is put in by hand, even the positivity of the energy-momentum operator is an axiom in AQFT), but a mass gap is considered to be an essential feature of a full fledged rigorous theory of QCD, for reasons that are explained in the problem description of the Millenium Problem of the Clay Institute that you can find here:


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*Yang-Mills and Mass Gap
A: I think it's beause do not have a fundamental understanding of mass. If we did, maybe that fundamental unit would have some relationship (i.e. tiny fraction) with the Planck mass.
The current effort in that direction probably begins with understanding the Higgs. There are several competing theories of the Higgs. They don't even agree on the number of such particles. So in that sense, the ball is in the experimentalist court.
A: The coupling constant for gauge theories is dimensionless, such as the fine structure constant $\alpha~=~e^2/(4\pi\epsilon_0\hbar c)$ $\simeq~1/137$.  Mass has naturalized units of reciprocal length.  This makes the establishment of a charge more reasonable, and a unitless number is something which is benchmarked as being an absolute constant.  In other words, if $\alpha$ changed it would be a pure numerical variation.  Occasionally there are claims of this.  A quantity which has an actual dimension in units is so in relationship to other quantities.  
This is a question related to the problem of quantum gravity.  The Planck mass $m_p~=~\sqrt{\hbar c/G}$ can be thought of as the fundamental unit of reciprocal length, and the gravitational constant $G$ has units of area.  This area corresponds to the unit area of a black hole event horizon.  For a Yang-Mill field theory the coupling constant functions in a field which is unitary.  By contrast units of mass are related to this reciprocal length, which in turn is not just a unit involving gravitational modes, but also the degeneracy of modes which have an entropy --- or entanglement entropy.
So mass does not quantize in quite the elementary fashion we might expects with charge and other coupling parameters for interactions.
A: Charge comes from discrete symmetries and is  countable and additive.
Mass comes from continuous 4d space,  is exchangeable with energy and, in quantum mechanical dimensions not linearly additive, thus not countable.
Suppose you have an elementary quantum of mass, $m_q$. In the world we know two such quanta would not end up as $2m_q$. 
One would add the four vectors and take the measure in 4space, and square root it, to get the invariant mass of two of them, etc for higher numbers at will. Given a mass, you could never know/count of how many $m_q$ it is composed. It is a continuum. Whereas charge is simply additive and countable.
The only way an elementary particle rest masses could be a linear sum of $m_q$s is for there to be no binding energy, and experiments tell us the elementary particles are bound, if stable.  If there were no binding energy then the composites would crumble into the constituent $m_q$ with the slightest scattering.
A: Dear asmailer, the reason is simple and completely understood: the electric charge is the generator of a $U(1)$ symmetry which is compact and may be parameterized by an angle, $\phi$. So wave functions may only depend on the angle $\phi$ in a periodic way, $\exp(iQ\phi)$ where $Q$ is integer (or an integer multiple of $e/3$, if I look at the elementary $U(1)$ rescaled by a factor of three that also allows quarks).
On the other hand, the mass is nothing else than the energy measured in the rest frame. The energy generates translations in time - and time is noncompact. So the corresponding phase $\exp(Et/i\hbar)$ isn't constrained by any condition of periodicity. So the energy is continuous even in the rest frame.
In the other frames, the continuous character of the energy is even more obvious because the "already continuous" rest mass is multiplied by the Lorentz factor $1/\sqrt{1-v^2/c^2}$ which changes - and has to change - continuously as we vary the velocity; the latter is required by the principle of relativity. So the mass and energy are continuous, have to be continuous, and will always remain continuous.
You could continue to ask "why" and in fact, you could get even deeper answers. You could ask why time is not periodic - which was used for the continuity of energy in a particular frame. Well, time has to be "aperiodic" because a periodic time would cause the grandfather paradox and other bad things - closed time-like curves. Time is also unbounded in the future because we live in a space with the positive cosmological constant.
On the other hand, groups such as $U(1)$ have to be compact and are compact in any quantum theory of gravity. This was argued e.g. by Cumrun Vafa in his Swampland program. For $U(1)$, the situation is simpler: the electric charge has to be quantized because of the Dirac quantization rule and because of the existence of the magnetic monopoles which is also guaranteed in a consistent theory of quantum gravity as was explained in another question on this server.
A: Mass is determined by how a particle interacts with the Higgs boson(s). Mass is also
determined by the relativistic mass-energy equation $E^2=m^2c^4+p^2c^2$ or more simply $m=\sqrt{(E^2-(pc)^2)}/c^2$  The energy values are a continuum, so there is no discrete elementary mass unit. In General Relativity, things are more complicated. In Stationary spacetimes, for example, the gravitational potentials (metrics) are not functions of time and the ST has time translational symmetry, so energy is conserved-- but while the stress-energy tensor is Lorentz covariant, in a non-isolated system, the system exchanges energy-momentum with its environment and its "mass" is not invariant--once again, no elementary or fundamental mass.
I hope this is what you meant, but maybe you just meant the Planck mass...
Frank Wilczek spends quite awhile in his popular book "The Lightness of Being" trying to say what mass is and isn't. He does quite a good job in non-technical terms-
http://www.amazon.com/Lightness-Being-Ether-Unification-Forces/dp/B004HEXSXG/ref=sr_1_1?s=books&ie=UTF8&qid=1296458301&sr=1-1
