Your question leads to many others...
--Would a different geometry such as toroid work better?
--What exactly does "about this size" mean? That is, do you include only the coil, or the volume out to where the magnetic field is below some value, for a given tank stored energy?
--A related question might be, what's the best coil Q for a given shield enclosed volume?
--What is the point of diminished returns such that adding more layers is not worth the effort?
--How much difference is there between winding several layers insulated from one another, and using an edge-wound tape wire (rectangular- cross-section wire, much taller than it is wide)?
--How accurately can you predict the Q improvement by simply taking into account the coupling among the layers, and ignoring proximity effects from layer to layer, and how does that change with frequency?
--Is there any advantage to using (slightly) different numbers of turns per layer?
---and many more
Many thanks to Sven for this thought-provoking posting, and the work he put into the multi-layer coil and its measurement.
Cheers, Tom