A formal system is defined as an abstract structure used for inferring theorems from axioms according to a set of rules (from Wikipedia).
Language, logic and mathematics are considered as formal systems. Is there any non-trivial non-formal system at all?
Science, economics, politics, almost anything I can imagine seems to fit into the definition of formal systems. They all rely on certain axioms that one could not prove. They all described by some form of language with symbols. In all of them we use inference rules to arrive conclusions and to answer specific questions.
Trivially, you can always construct some systems that are non-formal by simply discarding one of the features in the definition of a formal system. e.g. Take a piece of paper, write some finite number of definite symbols each identifying something but without any underlying inferable rule. Shape of your symbols can be considered as axioms, the symbols can form a very primitive language, but you can't derive any new theorem out of it because they are constructed without any rule. This might be an example of a non-formal system, but this is trivial. I am looking for an answer from the "real world". Can you provide any?
Natural science is not a formal system. We use formal systems to model the real world. It is not the real world itself. When a formal system no longer fits to describe the world, we discard the formal system (because we cannot discard the world).