It depends on the lipo. Anywhere from 1.x to 3mm thick in my experience. They're thinner along the seams. It's a fairly resilient plastic, a few of mine have been ejected from the car and dragged along the track or sent sliding along the asphalt/concrete surface with only cosmetic damage. (label torn off, hella scratches)

Avoid the lipos with the carbon fibre look case. Those tend to be thinner and more fragile, once again IME.