Here is an illustration of Gerry Myerson's nice idea:

![onions][1]

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][2]; doi:[10.1109/TIT.1985.1057060][3], [Semantic Scholar][4]. 

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][5], doi:[10.1016/0020-0190(95)00193-X][6].


  [1]: https://i.sstatic.net/mwqEA.jpg
  [2]: https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.113.8709 "zbMATH review at https://zbmath.org/0573.68035"
  [3]: https://doi.org/10.1109/TIT.1985.1057060
  [4]: https://api.semanticscholar.org/CorpusID:6738461
  [5]: https://www.sciencedirect.com/science/article/pii/002001909500193X "zbMATH review at https://zbmath.org/0900.68422"
  [6]: https://doi.org/10.1016/0020-0190(95)00193-X