I have one pair of metal frames with a 38mm lens width, and with a
-6.25 prescription, the CR39 lenses stick out at most only about 1.5mm
from the frame. The maximum total lens thickness is just over 4mm.
Besides the small lens, the thickness was also minimized by having the
optical center very close to the physical center of the lens.

On another metal pair with CR39 lenses, with the optical center of the
lens closer to the nose (which is very typical), and a higher -7.00
prescription and a 48mm lens width, the maximum thickness is about
7mm, with just under 4mm sticking out from the frame. The edges aren't
polished, so I would consider this about a worst-case scenario. It's
not that good, but it's not terrible, either. I get almost exactly the
same measurements on a frame with a 50mm lens and the same
prescription.

I see three main factors - lens width, location of lens center, and
edge polishing.

The fourth factor is lens material, but so far I was only looking at
CR39 lenses. On a metal frame with a 41mm "Thin and Light" 1.57 lens
and a -6.50 prescription, the maximum thickness is 4mm, and the lens
sticks out almost 2mm. The optical center of the lens is somewhat
toward the nose, and would be better centered if not for the wide 26mm
bridge on this frame.
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