Can a virtual image be magnified? It's common to encounter the diagrams of compound microscope describing its function. It consists of a real, magnified image formed by the objective and is further magnified by the eye piece. Can a compound microscope function wherein the objective creates a virtual but larger magnification and is further magnified by the eye piece? I feel my proposition might give rise to a larger magnification than the conventional methods. But I would like to know if my idea is physically feasible. 
 A: Your proposal is actually one of the main principles behind the design of high numerical aperture, high power microscope objectives, or at least it was until the widespread use of computer aided lens design. And it works even better than you might think. Indeed it is one of the few widely used perfect imaging systems to my knowledge. It is an elegant and beautiful idea.
This is the concept of the aplanatic sphere, sketched below:

Here we have a sphere of radius 1, of refractive index $n_2$ steeped in a medium of refractive index $n_1$. We consider a source at point $P$ inside the sphere, lying on the concentric sphere of radius $n_1/n_2$. You can, with a bit of geometry and Snell's law, show that the blue rays diverging leftwards from the sphere come from a virtual image at point $Q$, which lies on the concentric sphere of radius $n_2/n_1$. Moreover, the virtual image is perfect: all the rays, emerging leftwards from the sphere converge exactly on the point $Q$.
It gets even better: owing to spherical symmetry, the same reasoning holds for any point on the sphere of radius $n_1/n_2$: it has a perfect virtual image on the sphere of radius $n_2/n_1$. So the so-called aplanatic sphere is a virtual imaging device with a magnification of $n_2^2/n_1^2$ and which perfectly images a spherical surface to a spherical surface.
One now deploys this idea in a high power objective as sketched below. This image I have taken from the Zeiss Campus Website.

The aplanatic sphere is a hemisphere: through the use of immersion oil, the spherical object surface of radius $1/n$ inside the specimen is mapped to the virtual image $P(1)$ at radius $n$ by the aplanatic sphere. We then position a second meniscus lens above the aplanatic sphere such that its inner surface (closest to the specimen) is the same radius of curvature as the first virtual image surface through $P(1)$. The rays are thus undeviated on entering the second meniscus lens. The outer surface then forms a second aplanatic sphere: it is concentric with the distal end hemisphere and it maps the first virtual image $P(1)$ to the second virtual image $P(2)$. Again, the imaging is perfect. Now the magnification is $n^2$.
One can continue this process a number of times, stacking meniscus lenses fulfilling the aplanatic condition until the still perfect virtual imaging surface is far off in the distance, and the magnification is $n^M$, where the aplanatic sphere trick has been repeated $M$ times. The rays are near to collimated: it is now a simple matter to collimate them perfectly without aberration.
Before the widespread use of computerized lens design, this was the main way microscope objectives with a wide field of view were realized. Nowadays, one can have an optimizer evolve a highly corrected, wide field of view system. It's interesting that, if one gives the optimizer only criteria defining a wide field of view with low aberration and begins with a guess system comprising plane slabs of glass, a genetic optimization software will tend to come up with the classic aplanatic sphere design. I don't particularly like the result as a designer for the applications I am interested in because the design tends to leave you with thin meniscus lenses that are hard to mount accurately. With a computerized system at your disposal, it's often better to sacrifice some efficiency (i.e. allow one or two more surfaces) to achieve the same result with fatter lenses that can be mounted more simply in an easier to machine housing and which will give the same performance with coarser mechanical tolerances. You end up with a system that is much easier to build.
