I believe the lens material matter on how dark it can be tinted.
Something to do with how compatible is between the hard coat and the
lens, then the compatibility between hard coat and the tint. Then you
compatibility have between tint and ARC, then ARC and the teflon (or
something similar).
Polarization has nothing to do with how dark the sunglasses get.
Polarization causes light to pass thru the lenses in a certain ways,
such that you can see better against the reflective surfaces, like
water. Polarization also greatly reduce or eliminate reflection.
I have read somwhere that CR-39 and glass are the two lens materials
that can be tinted very dark. I am not sure about the mid and high
index lenses. Maybe Trivex can be also be tinted very dark but I have
only come across very few articles about it.
Many of the brand names non-prescription sunglasses have CR-39, but
the lenses are plano so they are quite thin compared to prescription
sunglasses with CR-39 lenses.
I have not found online stores that would do polarized prescription
sunglasses for cheap. The least expensive I saw was around $70. And
that is just the cost of the lenses.