Could you use a 3D printer to make a 3D printer's parts? What is the most of a 3D printer that can be made this way? Could parts that break be replaced this way?
Yes, you can print most of the parts (electronics, linear guide rails, ball bearings and nuts and bolts, etc cannot be printed). Actually this was exactly the purpose of RepRap.org:
RepRap is humanity's first general-purpose self-replicating manufacturing machine.
Since many parts of RepRap are made from plastic and RepRap prints those parts, RepRap self-replicates by making a kit of itself - a kit that anyone can assemble given time and materials.
I have built 2 custom printers myself using other printers to print parts and printed all printer parts for several others. It is possible to print your own linear bearings from POM, I prefer these over the noisy metal bearings.
Can a 3D printer make a 3D printer?
Not without a bunch of other parts
Could you use a 3D printer to make 3D printers parts?
To some extent.
What is the most of a 3D printer that can be built this way
Basically you you can't print the precision mechanical guides, you can't print the electronics, you can't print the parts that need to get hot, you can't print the motors, you can't print the lead-screws or toothed belts. Printing the whole main frame doesn't seem to be very practical either.
Afaict what you are left with is mostly printing the pieces that join all the functional elements of the printer together.
Yes, printers can print printer parts. With the right design, all non-metal/non-electronic parts can be printed on an FDM printer, and a laser-sintering based printer (SLS/DMLS) can even produce metal parts, as long as they fit the printbed. The fact, that a printer can create all the fastenings to build a new printer with just the electronics and linger extrusions/rods/rails is actually the core of the RepRap philosophy.
Some early printers, like the Mendel, did use just materials from home depot and printed joiners together with the electronics.
As a result, all joiners can be printed as replacements or improvements. There is a second limit though: Very small printers might not be able to print parts of the size needed to replicate the parts that they are made from. A common 200x200x200 mm printer can replicate almost all fasteners that comprise it, but the frame will have to be made from sheets of metal, wood, extrusions or rods and the movement system will need rails, rods or extrusions.
You can print some of it but not all, as 3d printers can't print metal or silicon to make the boards and frame.