The limit can be within $\varepsilon$ of any convex polytope: 

Start with a polytope $P$ then replace each vertex $v$ by three vertices $v_1,v_2,v_3$ forming a   triangle with sides of length (less  than) $\varepsilon$ with centroid $v.$ and containing no point of $P$ other than $v.$ This gives a polytope $P_1 \supset P$ with three times as many vertices but with all points within $\varepsilon$ of (the closest point of) $P.$ Then as the edge midpoint process is applied to $P_1$ one will get a nested series of polytopes all containing $P.$ $P_1$ will have many faces aside from the tingy triangles and the centroids of these faces will be in the limit body, so the limit os strictly larger than $P.$

I'd still like to know the exact limit for a regular tetrahedron (or irregular, it shouldn't matter) and other simple cases.