Are we technically able to make, in hardware, arbitrarily large neural networks with current technology? If neurons and synapses can be implemented using transistors, what prevents us from creating arbitrarily large neural networks using the same methods with which GPUs are made?

In essence, we have seen how extraordinarily well virtual neural networks implemented on sequential processors work (even GPUs are sequential machines, but with huge amounts of cores). 

One can imagine that using GPU design principles - which is basically to have thousands of programmable processing units that work in parallel - we could make much simpler "neuron processing units" and put millions or billions of those NPUs in a single big chip. They would have their own memory (for storing weights) and be connected to a few hundred other neurons by sharing a bus. They could have a frequency of for example 20 Hz, which would allow them to share a data bus with many other neurons.

Obviously, there are some electrical engineering challenges here, but it seems to me that all big tech companies should be exploring this route by now.

Many AI researchers say that superintelligence is coming around the year 2045. I believe that their reasoning is based on Moore's law and the number of neurons we are able to implement in software running on the fastest computers we have.

But the fact is, we today are making silicon chips with billions of transistors on them. SPARK M7 has 10 billion transistors.

If implementing a (non-programmable) neuron and a few hundred synapses for it requires for example 100 000 transistors, then we can make a neural network in hardware that emulates 100 000 neurons.

If we design such a chip so that we can simply make it physically bigger if we want more neurons, then it seems to me that arbitrarily large neural networks are simply a budget question.

Are we technically able to make, in hardware, arbitrarily large neural networks with current technology?

Remember: I am NOT asking if such a network will in fact be very intelligent. I am merely asking if we can factually make arbitrarily large, highly interconnected neural networks, if we decide to pay Intel to do this? 

The implication is that on the day some scientist is able to create general intelligence in software, we can use our hardware capabilities to grow this general intelligence to human levels and beyond.
The approach you describe is called neuromorphic computing and it's quite a busy field. 

IBM's TrueNorth even has spiking neurons. 

The main problem with these projects is that nobody quite knows what to do with them yet. 

These projects don't try to create chips that are optimised to run a neural network. That would certainly be possible, but the expensive part is the training not the running of neural networks. And for the training you need huge matrix multiplications, something GPUs are very good at already. (Google's TPU would be a chip optimised to run NNs.)

To do research on algorithms that might be implemented in the brain (we hardly know anything about that) you need flexibility, something these chips don't have. Also, the engineering challenge likely lies in providing a lot of synapses, just compare the average number of synapses per neuron of TrueNorth, 256, and the brain, 10,000.

So, you could create a chip designed after some neural architecture and it would be faster, more efficient, etc …, but to do that you'll need to know which architecture works first. We know that deep learning works, so google uses custom made hardware to run their applications and I could certainly imagine custom made deep learning hardware coming to a smartphone near you in the future. To create a neuromorphic chip for strong AI you'd need to develop strong AI first.