Are dimensionless physical constants predicted to be rational, irrational, or transcendental numbers? Are dimensionless physical constants predicted to be rational, irrational, or transcendental numbers?
Directly measured ones are obviously unknown, but according to Wikipedia many dimensionless constants have been computed from theory. I'm interested in replies from any class of theory.
 A: It depends on the constant you are talking about. For instance, it is an observational fact that the dimensionless constants $q_1 / q_2$, where $q_1$ and $q_2$ are the electric charges of two arbitrary particles, are rational numbers. It is a longstanding question of theoretical physics to understand why it is so. 
In some cases, one can explain why constants are indeed rational. For instance, if you are talking about charges with respect to a non-commutative gauge group, then the general theory of representations of Lie algebras shows that ratios of charges have to be rational. One can not directly apply this argument to the problem of the previous paragraph, because the gauge group of electromagnetism is $U(1)$, which is commutative (we also say "abelian"). 
On the other hand, to my knowledge nobody knows anything about ratios like $m_p/m_e$, where $m_p$ is the mass of the proton and $m_e$ is the mass of the electron. From mearurements we know that this can't be a rational number with a small denominator, but nothing more. 
A last remark : I'm not sure many physical constants (if any) have been computed "from theory", if by "theory" you mean "string theory", as your tag may imply. 
A: “With the development of quantum chemistry in the 20th century, however, a vast number of previously inexplicable dimensionless physical constants were successfully computed from theory.” I guess that you ask this question because you read the above sentence. But quantum chemistry is not the fundamental theory, it is based on quantum mechanics. So "dimensionless physical constants" mentioned in this sentence must be some combination of  constants in quantum mechanics, such as $\hbar,$ $e$ etc.
String theory and Grand unified theory(GUT) may be promising to derive the dimensionless physical constants. But both of the theories depend on some fundamental scale(string coupling constant or GUT scale). It is meaningless to ask the dimensionless physical constants are rational or not, because they must depend on another constant to be measured by experiment.
In addition, physics is not math. You can never prove the sun will rise tomorrow by mathematical theorems, although you can predict correctly by physical laws. Math has nothing to do with the world but physics depends on our detection of the world.
A: Physics does not predict that dimensionless constants will be anything. That would mean that nature favours one specific kind of numbers over the others. However, as  user40085 indicates above, some ratios are found rational. This is probably hinting to some unknown symmetry principle fixing that ratio. 
On the other hand, more "fundamental" constants such as the Newton Constant for instance are probably just set by nature. I would expect these constants to be irrational numbers, just because there are more of them. That is not a prediction,however.
About string theory fixing constants. It has the power to fix all of them but the string length. We don't know how to do it at the present stage.
