If we dropped an object into the Pacific Ocean, would it completely sink to the bottom of the ocean floor or not? Obviously if the density of the object is smaller than the density of water it would not sink at all. But I was thinking that, even if the density of the object is greater than the density of water, maybe the object would not sink to the bottom of the ocean floor because the pressure of the water increases as the depth increases. So maybe after a certain depth the pressure of the water would become great enough to counterbalance gravity and push back the object towards the surface.
And if the object would not completely sink to the bottom of the ocean floor, at what depth would the object stop sinking and stabilize itself (depending on its density)?
 A: If everything is incompressible then pressure does not affect buoyancy. 
If the object is less compressible than the fluid then the object will get more buoyant as the pressure increases. Conversely if the object is more compressible than the fluid then the object will get less buoyant as pressure increases.
Temperature and salinity can also vary the density of the water. Water at depth is likely to be a bit more dense.
So if you have an object that is pretty damn incompressible and is only slightly more dense than the surface water then it might find a depth and sit there. In practice few objects are likely to fall into this category.
A: In a static situation, the water would have a larger density when it has a larger pressure, so you could sink until you reach a level where the density matches.  In reality, the water can have a different temperature and a different salinity (both of which affect density) and if can be flowing (up/down, east/west, north/south) and so it might never settle down to a specific depth depending on what is going on.
A: It's not the absolute pressure that pushes a buoyant object toward the surface:  Pressure in a fluid exerts force in all directions.  It presses in (i.e., up) on the bottom of an immersed object, it presses in (i.e., down) on the top of the object, and it presses in on the sides.
Buoyancy happens because of the difference in pressure between the top and the bottom of the object:  Water exerts a greater force on the bottom of an immersed object because the bottom of the object is at a greater depth than the top.  That difference doesn't change (i.e., the net buoyant force doesn't change) as the whole object sinks or rises unless;
a) The density of the water changes, or
b) The object itself is compressed or expanded, thereby changing the difference in depth between its top and bottom.
Water is extremely stiff:  It's density changes very little with depth.  It's so stiff that in most practical problems, we call it "incompressible".  @PeterGreen's answer explains why the incompressibility of water means that virtually all objects will either float to the surface, or sink all the way to the bottom.
A: I was told that a human body will naturally float. But, if its taken down to a depth of $100$ ft (not sure if its fresh or salt water) the air in the body compresses to a point where the body is more dense and heavier than the volume of water it displaces, so it will stay submerged at that point. Raising it back up lets the air expand and restores its buoyancy.
I have a NAUI open water one certification. Whenever I ascend I never go faster than the smallest bubbles. And I have to keep choosing a new bubble to follow every few seconds because every bubble expands as it rises, making it rise faster exponentially. You do not want the oxygen in your blood to expand too quickly.
