In other words, what products synthesized by ribosomes are actual parts of ribosomes (if any)? How are they involved in their own synthesis otherwise? What is the cycle/chain of products starting with the proteins synthesized by ribosomes and actually ending with the synthesis of a ribosome?
The most important parts of the ribosome are not made by other ribosomes - 5 rRNA (ribosomal RNA) of the ribosome actually do most of the direct work of creating the protein and are made by RNA polymerase ( a protein, but not the ribosome).
Then there are 92 ribosomal proteins, which as a rule bind to ribosomal RNA to support their structure and keep everything going. These are all made by ribosomes. They are thought to have appeared later in the evolution of the ribosome though I imagine that it would not be possible to constitute a working ribosome without each one of them.
these numbers are for the eukaryotic ribosome, the prokaryotic ribosome has 3 rRNA and 52 ribosomal protein components.