What if the Planck constant was exactly zero?
I was thinking of a parallel universe in which the Planck constant has an exact value of zero. How would this effect the physics of that universe and how would it effect the development of that universe?
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Planck's constant is the quantum of action (energy transmission), so basically, this would be a world in which energy is not quantized, but rather infinitely divisible. This breaks our contemporary understanding of physics so badly it's hard to say what features a universe like this might have. If such a universe managed to have particles at all, they wouldn't be particles familiar to us, and their behavior would be completely different. Throw out your chemistry books... don't even count on having electrons.
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This question really has no answer because it is phrased in a way which doesn't actually have much meaning.
As phrased, it appears as though the universe is governed by these constants, which can be dialed this way or that by some supreme force. In this universe, we'll set Planck's constant to $6.626070040(81) \cdot 10^{-34}\;\text{J}\cdot\text{s}$, in another universe, $0\;\text{J}\cdot\text{s}$. A more meaningful approach would be to say that the universe appears to be governed by some laws, and several of those laws happen to involve ratios that are expressed as a constant. This is important because it points out a key issue with the question: there is not just one way to set Planck's constant to $0$. The mere definition of a parallel universe which is identical to ours but with $h = 0\;\text{J}\cdot\text{s}$ is insufficient to describe the parallel universe unless it is massively degenerate.
If we were to presume that the universe was nothing more than a set of equations and Planck's constant is something you can just dial down to 0, we get some strange behaviors. The first is the one that SudoSedWinifred mentioned. There is no quantization of energy in this universe. All of quantum physics immediately degenerates in messy piles of broken equations. It goes downhill from there. I don't know enough QM equations by memory, but the end result of setting Planck's constant to $0$ may cause a divide by zero in quantum mechanics, which means the universe you describe simply cannot possibly exist. Even if that isn't the case, by the Planck-Einstein equation, $E = h\cdot\nu$, the energy of a photon is proportional to its frequency, and the constant of proportionality is Planck's constant. If Planck's constant is $0$, this equation falls apart... photons can no longer carry energy. Not only is all of quantum mechanics completely and utterly torn apart, but electromagnetics also falls apart.
So the real question is not what happens when Planck's constant is $0$, but rather what you did to make the concepts behind Planck's constant lose meaning. You could state "energy has no quantization in my parallel universe," and build up behaviors from there. However, there is a difference between declaring the Einstein-Planck equation is $E = 0\cdot\nu$ and declaring that that equation is simply invalid and does not apply to this universe. You could also make a world where photons are incapable of carrying energy, and explore the consequences of that. In such a case, Planck energy is still a valid effect, and QM still (sort of) hobbles along, but there is simply no connection between those effects and photons.
Or you can just say "it's a parallel universe where $h = 0\;\text{J}\cdot\text{s}$," but the resulting universe is so degenerate that it wont be much fun to try to explore. It is as meaningful as "writing something on the other side of a Möbius strip." It sounds great, but doesn't actually mean anything (because a Möbius strip simply only has one side).
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Everything might be pitch black. Because there would be an infinite number of orbitals in each molecule (instead of discretized solutions to the Schrodinger equation (i.e. Hamiltonian) you would have infinitely many solutions to the spectra at infinitesimally-different energy levels) any photon could be absorbed by any molecule (unless some other weird effect stopped it...). Thus all matter would absorb all wavelengths of light.
Another problem would be the blackbody radiation paradox, otherwise known as ultraviolet catastrophe. Without quantization of photon energy, energy would be radiated at infinite energy at all temperatures.
Luckily the infinite energy would be immediately absorbed by the infinitely absorbing matter around it. :)
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