Is each cube wall to wall neurons?

While working on a cube, it looks really packed with neuronal connections. How do cells move their dendrites and axons around when it looks like there’s no room to move? Do they have to push between walls of neurons to make new connections? Is everything in the cube neurons or is there other stuff?

I was always under the impression that dendrites were near static, they might die and regrow(?), but not move around. Now I’m curious about the answer from knowledgeable individual.

Neurons do move. Not in a sense that they float around as red blood cells do in blood vessel, but in a sense that they make new branches and stretch them to make new synaptic connections or disconnect existing synapses and degenerate branches. The process is called the neuroplasticity and it is the very basis of learning and memory. 

There are a few facts to consider about the plasticity in the retina and about your question. 
1) In the cubes, there are non-neuronal cells called as glia, but they are also a part of neural system. Sometimes you may also encounter blood vessel with thick swirling wall. But you're right. As you pointed out, the neurons are very tightly packed to each other. On the other hand, the neurons we are looking at were blown up a little bit during the sample preparation, making the extracellular space a little smaller than when it was in the living mouse. Nevertheless, the effect must not be critical.
2) The plasticity in retina is not as strong as that of the rest of the brain. It is believed that the retinal neural network is mainly designed by the genes. Each neuron is told by the genes what to do, where to go, and which other neurons to connect to. An organism processes the visual information in almost the same way as other individuals of its kind do. 
3) So let's talk about the plasticity in general. While scientists know the cellular and molecular mechanism of the process pretty well, and they can even observe a few neurons change their shapes and connections on a petri dish, I'm not sure if anyone has ever seen this in a bulk tissue simply because it would be technically difficult. To do this, you will have to find a way to "look through" the bulk and will have to know which neuron to look at. 

So to tell the truth, we do not know so much about what really happens during the process of plasticity. Given the circumstances, it wouldn't be too wrong to assume that a growing branch of a neuron must crawl its way through the jungle of other neurons, but the final answer is yet to be studied.

I recall reading a paper that examined synapse formation in the motor cortex over the course of a mouse learning to perform a new motor task. I think it was published by a lab at UC Davis, Berkeley, or San Diego, but I may be wrong. I will try digging it up for you.

Edit: Here’s the article I was thinking of. It doesn’t deal with dendrite growth and formation but it does examine how motor task learning impacts dendritic spine formation and elimination the motor cortex. Also, the lab is at UCSC