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I really love small neural networks. They have some nice properties that people overlook. The training speed record (warning, self promo) for CIFAR10 to 94% uses a very tiny neural network (<10 MB if just saved raw out to disk as a definition file). That's located at https://github.com/tysam-code/hlb-CIFAR10.
You could make that even smaller if you wanted to, though at least this network is already pushing maybe even a little further down the diminishing returns spectrum in some areas than I'd like.
I think a really fun challenge would be to find the fastest network that infers at 94% in under 1 MB. I certainly believe it's possible, but with pareto laws the way they are, it would take a whole lot longer to train and might not be as fast on a GPU during inference as the main net (despite having fewer parameters). That might not be true, however.
There's a few NP-hard problems that actually exist in this space that not a lot of people talk about but I feel will be considered a core part of the theory of training neural networks at some point in the future. The size of the network is a very interesting tradeoff that opens up certain mathematically interesting properties on either end of the spectrum. Bigger is not always better, though it is simpler and simple oftentimes survives.
One of the common threads (might be a "common", I'm not sure to be honest as I live in my own personal bubble of research interests and community and etc) is the dimensionality of the problem at hand. That plays into the scale of the network used to solve a problem. I remember some discussion being sparked a while back from some Uber research about the inherent dimensionality of a neural network on a particular problem (though of course it's naturally linked to your inductive bias so please take that as you will). As you noted, some networks do quite well with very few neurons, 15 is a record however from what I've heard (and I'd love to see that -- I have a guess as to which particular method, or, at least, method family, it is... ;P I'm...casually interested in that arena of research).
In any case, as you can see I am quite interested and passionate about this topic and am happy to discuss it at length further.