If you dig a bit deeper into skin effect you will find that the often quoted number is "from each surface". A wire of 16 mm diameter has zero area that is outside one "skin" deep. The current does not go to zero at that point, it simply drops off a bit. As you go into a wire, the main part of the cross section is towards the edge rather than the center (go in 1 mm from the outer edge of a 10 mm wire, that's a lot more area than the center 1 mm radius section). The cross section effect means that the practical impact on a cable is quite small even in the 16 to 20 mm range.

Close, but not quite!

The actual condition is for a half-infinite conductor. Thus, the surface is an infinite plane, and there is unlimited depth behind it (current drops off exponentially with depth from the surface, never quite reaching zero, but getting close enough for engineering purposes by about 3 depths). The surface is perfectly uniform, no curvature and no interference from other currents.

An infinite plane is also the surface of a cylinder or sphere with infinite radius. If you shrink the radius, the surface becomes curved, but this still isn't very important until the radius is on the same length scale as the skin depth.

When the radius is less than 3 skin depths or so, you want to use a different analysis. The cylindrical case (for a lengthwise current down a solid cylindrical wire) ends up with a Bessel function instead of an exponential. Thus, there are in fact zeroes within the depth of the wire! This can be imagined by considering that the conductor necessarily exhibits attenuation and phase shift: as the current diffuses into the depth of the conductor, it is phase shifted, and therefore interferes with the waves diffusing in from other sides. Thus, there can be one or more shells where current goes to zero, and in fact reverses inbetween each!

As an approximation, the infinite-plane case is still a good estimate. For cylinders of radius under 1 skin depth, the current density is still pretty uniform (i.e., the radius is less than the first zero of whatever Bessel function shows up here).

When you have nearby currents, things get tremendously more complicated, with only a few simple cases having analytical solutions I think. You tend to get elliptical integrals from mirror-symmetrical and rectangular geometries, like twin-lead or PCB traces, which quickly get easier to compute by brute force (general EM field solver) than trying to work the equations.

You can still hand-wave your way through things, so for example, a thin strip will have most of the current flow along the edges of the strip, and little in the center. The skin depth is working in from the edges, even though the thickness of the strip is much less than it. So it's "edge depth" rather than "skin depth", but the physics is identical. (This can be treated as a 2D diffusion problem, where the strip is thin enough to consider as a plane conductor with resistance/sq instead of bulk resistivity. A thinner strip has deeper edge effect, because its area resistivity is higher.)

Which pretty quickly gives rise to being able to foresee and describe (if not compute) the edge effect, current crowding and proximity effect of windings in a transformer or inductor. The effects are all scaled by the same frequency dependence, but the multipliers can be quite large, 10 or 20 times worse for wires deep in a winding or Litz bundle!

Tim