Before we move on to your question, I’ll just mention that the referenced well in the OP is a world record for horizontal drilling, but not vertical drilling. In this case, the 6,421 figure represents the total measured depth of the borehole, including 184m of horizontal wellbore. The total vertical depth (TVD), measured from surface, was 6,062 meters (19,888ft). There are many wells with much longer horizontal sections, and numerous wells drilled for oil and gas have exceeded this vertical depth. I personally have worked on a well that was drilled to >23,000 feet TVD.
The deepest well of any kind drilled to date was the Russian Kola-1 geologic exploration well, which reached 12,263m (40,233) in 1990. A German consortium known as KTB executed a geologic exploration well in Bavaria in the early 90’s that was originally intended to exceed this depth, but IIRC, was abandoned just short of 12,000m vertical depth due to drilling problems.
Now, on to your questions. All deep wells have been drilled via adaptations of standard rotary drilling techniques used in the oil and gas industry. This involves the use of a drill bit attached to a hollow, jointed drill string through which a drilling fluid is circulated. The drilling fluid a) cools the drilling bit; b) balances fluid pressures in the formations being drilled; c) provides a way to clear cuttings from the bit face and bring them to surface. The drill bit can be rotated from surface by turning the entire drill string or via a fluid-operated motor near the bit.
20,000 meters of vertical depth could be attempted using rotary drilling technology, but probably only in certain locations where the surrounding rocks are relatively stable. Depth-limiting factors would include:
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resistance of the drill string to rotary and bending forces;
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bottomhole temperature; temps above about 300 deg. C would tend to break down the drilling fluid and place excessive thermal stress on the drill string
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rock strength; loads produced by fluids in the pore spaces, or tectonic forces, would make the borehole wall unstable at some limiting depth, no matter what density drilling fluid is used.
Drilling in a deep ocean trench would pretty much be a non-starter, IMO, as the these are actually subduction zones that would feature very high temperatures and unstable rocks. The presence of the seawater column would present numerous engineering complications as well.
I know of no sedimentary basins that are estimated to reach 20,000m of vertical depth, so any scientific knowledge recovered would involve information about the structure of deeply buried igneous rocks (primarily from the cuttings), and such things as subsurface temperature gradients. While there has been some speculation about methane (natural gas) perhaps being brought up from deeper in the crust, IMO most experts in the oil industry do not endorse this view, and production of gas from igneous rocks (which are inherently non-porous) would be uneconomic unless extensive fracturing were present.
Hope this helps.