How deep could the surface of the Earth's crust get?
Challenger Deep (10.99km) in the Mariana's Trench is the deepest part of the ocean floor. This is probably close to the deepest theoretically possible ocean floor with the  current thermal regime.

With the current thermal regime, the deepest 'steady state' ocean floor depth is about 5.5-6km ( cooling curves at Muller et al ) - limit is reached as the ocean crust cools and subsides. The crust (lithosphere) is effectively floating on the mantle (asthenosphere), and this steady state situation is known as 'isostatic equilibrium'.

You can get deeper by moving from a static to a dynamic situation. A subduction zone provides this - the subducting plate is moving and being pushed down, resulting in a deep ocean trench. Eg. The Mariana's, or the trench offshore Chile. A fast subduction zone should go deeper. A deep hole like this will also tend to fill with sediment. The fastest subduction zones move at about 16cm (and these are only the short arcs). The physical characteristics of the ocean crust will also vary a little (thickness, temperature), but beyond these parameters it is difficult to make the hole deeper, as it were.