An Apollonian Gasket is a fractal set constructed out of tangent circles. The first stage is three mutually tangent circles (which are not all tangent at a single point). At each step, we can take a triplet of mutually tangent circles and construct two new, non-intersecting circles which are tangent to each circle in the triplet (this is Descartes' Theorem). Continuing in this way, we generate the fractal set.
Circles, of course, are remarkably symmetrical curves. It seems reasonable to ask if we can carry out this construction with other closed, bounded algebraic plane curves. For example, if we started with three mutually tangent ellipses with major axes twice the length of there minor axes (that is, of the form $$x^2 + \frac{y^2}{4} = 1$$ after some appropriate translation, dilation, and rotation), could we repeat the process? More generally, given a plane curve $C$, suppose we take three copies of $C$ and, using rotations, translations, dilations, and reflections, arrange them to be mutually tangent. Under what conditions (on $C$ and the initial configuration) can we repeated take triplets of tangent curves $C_1, C_2, C_3$ and construct some number of curves $C'_1, C'_2, \cdots C'_n$ which are transformations of the original curve $C$, non-intersecting, and tangent to each of the curves $C_1, C_2, C_3$?
Since an ellipse is an affine image of a circle, certainly you always can set up an Apollonian type packing with ellipses simply by applying an affine transformation to a circle packing. Here's a circle packing and the corresponding packing by ellipses that are twice as wide as they are tall.
The real issue involves uniqueness. That is, given three mutually tangent, non overlapping circles, there is a unique circle that is tangent to all three and not containing all three. Given three mutually tangent ellipses, there are infinitely many possibilities for the third ellipse, taking rotation into account.
Perhaps you could minimize the area of that fourth ellipse? Or maximize it, or work with it's perimeter, or fix its orientation, or any number of things.