A variant of Robert Israel's argument: Take a convex body $K$. Slice it by planes, to make a polyhedron $K_\epsilon$ whose Hausdorff distance to $K$ is less than $\epsilon^2$ and the diameter of faces is less than $C\epsilon$. Apply the edge-midpoint process. Then the limit body has many points at distance $<\epsilon^2$ to $\partial K$; one per face. Thus the set of limit bodies is dense in the set of convex bodies. In particular, most limit bodies are not ellipsoids.

A variant of Robert Israel's argument: Take a convex body $K$. Slice it by planes, to make a polyhedron $K_\epsilon\subset K$ whose Hausdorff distance to $K$ is less than $\epsilon^2$ and the diameter of faces is less than $C\epsilon$. Apply the edge-midpoint process. Then the limit body has many points at distance $<\epsilon^2$ to $\partial K$; one per face. Thus the set of limit bodies is dense in the set of convex bodies. In particular, most limit bodies are not ellipsoids.

Alternately, you can form a polyhedron $K^\epsilon$ containing $K$, arbitrarily close to $K$, by taking finitely many tangent planes. Then again apply the edge-midpoint process to obtain a limit body close to $K$.