Drilling through drywall, hitting mystery object
As it turns out, the reason all holes in this wall were running into something beyond the drywall is soundproofing.

The back of the drywall is laminated with another wood-like material to reduce sound-transfer since another apartment shares this wall.  

It is safe to drill through, and beyond it is the normal insulation-filled void you would expect.

Can make studfinders have difficulties though.