Making a literal thousand-layer cake?

The other night some friends and I ordered a thousand layer cake for dessert in a restaurant. Sadly, the cake had only ten layers, just 1% of what we ordered.

We started talking about whether, with only commonly-available ingredients and kitchen appliances, there was a feasible way to actually construct a thousand layer cake. We figured out that if you could make a very thin crepe, you could make one thousand layers by folding it in half ten times. That, unfortunately, doesn't seem like it would work - it's virtually impossible to fold paper or aluminum foil this many times, so I doubt a crepe would survive. Plus, it would end up being very narrow and very tall.

We also thought about making something out of phyllo dough, but we weren't sure whether we would have any way to guarantee that the end result actually had one thousand layers or whether those layers would be distinct.

Assuming you have a typically-equipped, normally-stocked kitchen and a six-hour time limit, is there a feasible technique for producing a cake that is guaranteed to have at least one thousand distinct layers?