Why can't we have a wave of particles? I understand the nature of light can be complex and has extensive theories/experimental data. We hear light can be both a wave and particle, so why can't it be both, a wave of particles?
 A: In the non-quantum sense, most waves we experience in the everyday world are waves of particles. Ocean waves and sound waves are the transmission of energy through a bunch of particles/the medium
A: The problem is that light is not a wave and it's not a particle. Light is a quantum field, or at least that's our current best description of it.
Quantum fields can behave in ways that appear to be wave like, and they can also behave in ways that appear to be particle like, and this the origin of the claim that light is both a wave and a particle. It's more accurate to say that light can behave like both a wave and a particle.
Anyhow, as far as I know quantum fields can't behave like a wave and like a particle at the same time. Generally speaking they behave like waves when propagating and like particles when exchanging energy. So we can't have a wave of particles or a particle of waves. It's one or the other.
Incidentally, this applies to all particles and not just light.
A: If the wave function was a wave of particles, then you wouldn't get an interference pattern if you did a double slit experiment with one particle at a time going through the apparatus. But there are many experiments in which an interference pattern is observed when particles are sent through double slits one at a time.
The only satisfactory explanation of single particle interference that I am aware of is the many worlds interpretation of quantum mechanics as described by David Deutsch, see Chapter 2 of his book "The Fabric of Reality".
A: IT seems that wave phenomena (properties) can possibly be applied to atoms, and then molecules - ie. a bunch of particles (properties) in coherent form, here:

The quantum superchemistry of the out-coupled hybrid matter waves is
  studied by using the powerful tool of positive-P representation in
  quantum optics. We identify the stimulated eﬀects of the molecular
  output step and atomic revivals by steering the rf output couplings.
  The coherent conversion of atoms to molecules is robust, with the
  noise-induced molecular damping occurring near the total conversion.
  If these eﬀects are typical, then the quantum superchemistry in an
  atom laser can be tunable and potentially useful for, e.g., making a
  continuous molecule laser.

Also here: An article (not copy/pastable) regarding interferometry of molecules -only 9 pages, worth a read.
A: I think you need to read de broglies Noble lecture:    

I thus arrived at the following overall concept which guided my studies:
  for both matter and radiations, light in particular, it is necessary to introduce
  the corpuscle concept and the wave concept at the same time. In other words
  the existence of corpuscles accompanied by waves has to be assumed in all
  cases. However, since corpuscles and waves cannot be independent because,
  according to Bohr’s expression, they constitute two complementary forces
  of reality, it must be possible to establish a certain parallelism between the
  motion of a corpuscle and the propagation of the associated wave. The first
  objective to achieve had, therefore, to be to establish this correspondence.  

From this I believe you are right in saying that at the same time, each photon is accompained by waves. 
A: Light is neither particle nor wave. It's something different we haven't fully understood. When we try to find behavior of light assuming it's particle, light says affirmative. When we try to find behavior of light assuming it's wave, light says affirmative again. Means, light is behaving like particle and wave at the same time.
But, here your problem starts: Why can't we accept from behavior that light is both? Ok, natures of particle and wave are different, but why can't it be a wave of particles?
Answer:


*

*A wave of particles is disturbance in the medium. When you fire sound waves, you actually fire disturbance not air molecules. So, if light is reaching a dark place from a distance, it'd be dumb to assume Photons are already there. And, if there aren't any Photons already, a Wave of Photon isn't possible.

*Even if you fire one Photon, it'd spread out like wave (which you shouldn't accept with one particle) and make interference patterns with other one Photon.
A: A simple question about light is, "where is it?"
The wave model says that there's no single precise answer. It's spread out.
The particle model says that each photon has a place in space.
A model that assumes both at the same time doesn't have a good answer to this.
Modern physics of course has much more complicated models, but they are made possible through math. And then it turns out there are rather identical models for other types of waves and particles, so much that De Broglie assumed that all those waves were particles and vice versa, i.e. that you could use the same unified mathematical model. (Which still didn't answer exactly what that model should be, though)
