Let $P_i$ be a set of points in the plane and $P_{i+1}$ the corners of the Voronoi diagram of $P_i$. Start with some $P_1$ and iterate away. What happens? Can you choose a $P_1$ so the iteration goes into a stable loop? (If you play the game on a torus, $P_1=${some point} works, on a sphere $P_1=${a tetrahedron} works either.) It's relatively easy to see that if $P_i$ has exactly two inner points, $card[P_{i+1}]=card[P_i]$, but this is neither necessary nor sufficient  nothing tells you how many inner points $P_{i+1}$ has.

$\begingroup$ I think whenever you have a lattice you will have lattice, you get such a loop. But can you find a finite set where $P_{i+1} \geq P_i$? $\endgroup$– flawrJan 21 '18 at 20:52

1$\begingroup$ @flawr The Voronoi diagram of the seven points $\{(0, 0), (4, 0), (0, 4), (4, 4), (2, 2), (3, 2), (1, 2)\}$ has eight vertices. $\endgroup$– Martin EnderJan 21 '18 at 23:31

$\begingroup$ Related question (not fully answered): "Does this iterated Delaunay triangulation process always “explode”?" $\endgroup$– Joseph O'RourkeJan 22 '18 at 2:07

$\begingroup$ Surplus conjecture: Let P be set of all six point sets that are symmetric to the origin and have two interior points. It looks like vor(p) is in P whenever p is, but I yet have no idea about what happens to the distances in the long run. $\endgroup$– Hauke ReddmannJan 27 '18 at 20:19
Not answering your questions, but here are five iterations starting with $10$ random points:
^{ In the first frame, the $10$ points are clustered in the large central dot. }
^{ Subsequent frames alternate Voronoi sites (orange) and Voronoi vertices (blue). }
And here is the last frame enlarged:
^{ Enlarged view of 5th frame in sequence above. }
^{ In the next, 6th frame, the blue points would become the orange Voronoi sites. }
It appears to me, from limited empirical exploration, that, for random points uniformly distributed in a square, the number of points $P_{i+1}$ is very roughly $1.8 P_{i}$ with each iteration. So growing from $10$ points to ${\sim}100$ points after $4$ iterations.
In a Euclidean space, the set $P_{i+1}$ is contained in the interior of the convex hull of $P_i$. Thus the convex hull of $P_{i+1}$ is a proper subset of the convex hull of $P_i$, and there can be no cycle.
EDIT: no, sorry, this is false. If $P_i$ are the vertices of an obtuseangled triangle, then the (only) Voronoi vertex is the circumcenter which lies outside of the triangle. Still, I hope that there is some kind of "monotonicity" that prevents a cycle...

$\begingroup$ Did you reverse the inclusions? The Voronoi vertices extend beyond the hull of the sites. (Your point remains.) $\endgroup$ May 19 '18 at 11:03

1$\begingroup$ Actually, neither inclusion holds in general. Take the vertices of a triangle as the starting set. There is only one Voronoi vertex: the circumcenter. Now it depend on whether the triangle is acuteangled or not... I have edited my answer. $\endgroup$ May 19 '18 at 11:56