Why would you have to go deep? Why not just add increasing weight a non compressible float and see how it reacts to varying depth? Or did I not understand your question?

I wonder how many of your incompressible solids start off with a low enough initial density to make this possible.

Is this you just started reading "Going Postal" by the way? One of my favourites.

A.

The two things are not related, per se, for sinking. Your assertion does not make sense. Water, being incompressible, for the most part, has a lower density than almost all structural solids. Archimedes' Principle states, therefore, that such solid objects will sink, period. The solids with lower density simply float. To achieve neutral buoyancy requires something with sufficiently low enough net density that can't get crushed before it reaches its neutral buoyancy depth.

The bathyscaphe Trieste made a 10,911-m dive into the Marianas Trench in the 1960. But, that was only because it's two-man cabin was specifically designed to withstand the crush pressure at that depth, which is on the order of 110 MPa. Note that it achieved this by separating the buoyancy means from the life-support means, which allowed the manned cabin to be designed to be a sphere, while the buoyancy mechanism was filled with a lower density fluid that was resistant to compression, which protected its container from being crushed.

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This is about the density (not pressure) of water does or does not vary with depth?

Yes, I think that what the OP is saying is that water, not being perfectly incompressible, gets a little denser with depth. There are other materials which are more incompressible and their density increases more slowly with depth. I think the question is whether anybody knows of real life examples where the density/pressure curve of a substance crosses that of seawater in a way (at a credible depth) that would allow an object to settle itself stably at a predictable depth in mid water.

You can achieve something that looks similar by ballasting an object to be neutrally buoyant, then releasing it into a strong thermocline or halocline - imagine a submarine "riding a layer".

Perhaps one answer is that it's just too simplistic to assume that compressibility is the main thing that affects seawater's density as you change depth.

A.

The effects of temperature and salinity on water density are much greater than the effect of pressure. So, while what the OP suggest is theoretically possible in one particular set of circumstances, I'm not aware of this effect being practically useful.
There are instruments called density gradient columns that measure density by the depth where an object floats.

OK, theoretically, yes, but pointless; water's density in the bottom of the Marianas Trench is 5.4% higher than at the surface.

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