It is a well-known fact that all combinators can be derived from the two fundamental combinators K and S. It seems only natural to also ask whether there is a single universal combinator, but I can’t find much information on the topic. The only examples of universal combinators I can find are improper ones that refer to other combinators in its definition, such as Chris Barker’s U combinator. I also found a discussion about the same question, but no definitive answers are offered there.
Is there a proof of (non)existence of a proper universal combinator, one that can be defined purely in terms of its own variables?
You are looking for Craig's theorem, which states that that there is no single proper combinator that is a basis for combinatory logic.
(Craig's theorem is not any more general than that.)
One reference is A new proof for Craig's theorem by Patrick Bellot. I have not been able to find the entire paper online, only the abstract; if I find a better reference I will add it here.
(Unfortunately, it seems there is a much better-known Craig's theorem in mathematical logic which is completely unrelated. I have written to Bellot for details.)
Addendum 2016-03-09: The Bellot proof is only one page long, and is now available from JSTOR for free online viewing. The citation given by Bellot is:
but he says that this proof “was shown to be incomplete by André Chauvin”.