Place n/4 balls separated along a horizontal line L_{1}, and another n/4 along a parallel line L_{2} below, with each of the lower balls directly below an upper ball with their centers 1 unit apart. Thus each pair of balls overlap, their surfaces intersecting in a circle. Arrange a second set of n/4 pairs of intersecting balls along lines L_{3} and L_{4}, far from L_{1}/L_{2} and with all four lines parallel, and such that all circles of sphere intersections are coplanar. Now it is easy to see that a line tangent to two circles of intersection, one from the L_{1}/L_{2} group, one from the L_{3}/L_{4} group, is tangent to four balls. And there are Ω(n^{2}) such lines. (The same bound can be achieved with disjoint balls with a similar arrangement, but the analysis is slight more complex.)
The problem is also interesting if all balls are disjoint; it is not clear if disjointness affects the answer asymptotically.