I can't think of any law deeper than the Law of Non-contradiction; I consider questioning it to be more fundamental than the question of God's existence.
Does rationality (any and all rationality) /presuppose/ that the Law of Non-Contradiction is always going to be there? Can there, alternatively, be "true contradictions"?: I'm referring to logic gluts (a proposition that is both true and false) and gaps (a proposition that is neither true nor false) -- or ought we consider the consideration of these nothing more than the fanciful experimentation of an unrestrained imagination?
(n.b. The common strategy to attack the Law of Non-Contradiction is via. a challenge to The Law of the Excluded Middle, which states that a proposition is either "a" or "not-a", but not "both" nor "neither".)
It's hard to think of many good examples (paraconsistent/non-Classical logic is a very new development, so is not yet furnished with a wealth of palatable metaphors), but self-referential sentences may serve:
"This sentence is a lie" might be an example of a gappy proposition. How legitimate of a challenge is the Liar's Paradox to Classical (Aristotelian) logic? Does it break the shiny, perfect toy that we call the Law of Non-Contradiction? Are we left only with tautologies like "A=A"?
If you want more context, please refer to:
https://plato.stanford.edu/entries/dialetheism/(I've mentioned this in earlier threads ("Paraconsistency" and "Fuzzy Logic"), but I think the phraseology of this OP is different enough to warrant another treatment.)