Here is an illustration of Gerry Myerson's nice idea:
onions http://cs.smith.edu/%7Eorourke/MathOverflow/Onions.jpg
The left set has onion depth $n/3$, the right set, after small rotations, has depth 1.
Incidentally, there is an efficient algorithm to find the onion depth of a point set: $O( n \log n )$ for a set of $n$ points, established by Bernard Chazelle in the paper, "On the convex layers of a planar set," IEEE Transactions on Information Theory, 31: 509-517, 1985.
There also has been some work on the combinatorial structure of onion layers. A crude summary is: the structure is complex and not well understood. See "Onion polygonizations," Information Processing Letters Volume 57, Issue 3, 12 February 1996, Pages 165-173.