OK, I got to it this morning. I had a friend over in the morning for a wood carving session, and then was busy the rest of the day. It is indeed a veneer, ~.08" thick. Can't tell what the bottom face is, it just looks like some cheap white wood. No stain on it. The top is nailed to the carcass, no cleats. I could pry it off, remove the edge pieces and put on a new oak plywood top. I'll have to check at the place I buy wood from. I'm going to make is shiny with some paint thinner and see if I can just live with it the way it is. My neighbor has a chest (I think that's what she called it) dating from the late 1700s or early 1800s she's been trying sell. I'll have to take a look at it. She says it needs some fixing, and only wants $100 for it.

Steven- Random Orbital Nailer