Is there an inductor type that has the best inductance per unit volume?
For best Q, it is a cylindrical coil of (length) Diameter/Length=2

Turns will be spaced apart.

This is simply the optimum point. If you make it longer, then the same copper would add inductance faster by increasing D. If you make it shorter you lose mutual coupling faster. See. Note that one thing this graph tells you is that flat coils are not the solution. 

Note it is Q you want. Best inductance/volume, just means bazillions of turns of very fine wire, and unusable RF performance.

There are plenty of other constraints that can act.
 - Self resonance inter-turn / interlayer C becomes an issue with air core coils. Modern radios use ferrite to keep N down. Before ferrite you see basket weave, cotton covered wire, spaced turn coils to keep the C down. 
 - skin effect: litz wire is used in medium frequencies to reduce effective R

Internal capacitance can be thought of as C which is bypassing the L - and thus removing it. turn-turn C is limited effect, because dV between turns is low. But if you wind a second layer of windings, like a solenoid/relay/transformer, then dV between ends is very high, and so a lot more current flows through interlayer C, than inter-turn C. So multilayer coils are undesireable. (Before you say What about the little IF coils in radios - if the size is very small for the frequency, because you are using ferrite, then the C is too small to matter, and you can wind however suits you

When your constraint is size, and as radios became more complex, more coils = less space, it does become the constraint, then you have to choose non-optimum, like multilayer windings. Now you wind like a ball of string, with the windings criss-crossing to keep the wire gap high.

Perhaps another perspective is:

For a given size, you can keep adding layers. As you add each layer, the inductance will go up. At some point the self-C dominates, and the inductance starts to go down again.