Gödel first incompleteness theorem states that certain formal systems cannot be both consistent and complete at the same time. One could think this is easy to prove, by giving an example of a self-referential statement, for instance: "I am not provable". But the original proof is much more complicated:
It was proved by constructing a statement that indirectly referred to itself as 'This statement cannot be proved' - to be more precise, it says: 'The $i$-th proposition is not provable'.
By looking just at this sentence, it certainly isn't self-referential, but if we look at how all the propositions were numbered, we can see that $i$ is the number of the above proposition, so the self reference is not direct. Is it what makes the theorem so important and the reason why the proof is so complicated - the fact that statements containing no direct self-reference whatsoever might still refer to themselves indirectly?