What is the rate of expansion of an image as it aproaches the human eye?
The magnification of an object at distance $g$ from a thin lens with focal length $f$ is given as

$$\beta=\frac{f}{g-f}$$

which is generally negative for objects at distances greater than the focal length. This means the image is upside down.

The derivative of the magnification with respect to the object distance is

$$\frac{\partial \beta}{\partial g}=-\frac{f}{(g-f)^2}$$

So for constant $f$ the magnification diverges like $x^{-2}$ where $x=(g-f)$ is the difference between focal length and object distance.

So yes, the magnification grows in an accelerated way and even approaches infinity.

However, the relationship is only strictly true if the image at the image plane is sharp and the aperture is unlimited. So for your eye, at some point the image will become blurry and the magnification will grow in an asymptotic way that is much harder to describe.