In theory, how deep can a natural formation on the surface of the Earth's crust be? What are the limits to this depth?
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.
In theory, a new fissure could open up at any time, leading to a new depth. But for a long while now, the deepest known natural point is the Mariana Trench - just shy of 11km deep!
However, we have gone deeper!
In 1970, the Kola Superdeep Bore project had several boreholes drilled from a central hole. The deepest - SG-3, reached 12,262metres into the earth. The goal was to get to 15000m, but the problem is that the drill bits start melting in the heat down there - they need to operate in over 300 degrees Celsius!
About the limits: The largest depths on land are created by thinning of the lithosphere (Dead Sea rift as far as I know). The limit to this lowering is of course the inflow of the ocean (So the deepest point you can reach on dry land can only be slightly lower than the ocean water level.
In the ocean the deepest points are the places where the subducting plate pulls the lithosphere down into the astenosphere. The longer the attached subducting plate is (before it detaches), the stronger the force will be. It is probably difficult to say how deep a subduction trench can get because as it deepens it is also filled with sediments.