Is there a magma with this property?

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Does there exist a magma $(S,*)$ such that the only quasi-identities that $*$ satisfies are the trivial ones? And if so, can someone give me an explicit example of such a magma?

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Any quasi-identity only involves finitely many elements of a magma, so if it fails to hold identically it fails to hold in some finitely generated magma. So we can take e.g. the free product of all finitely generated magmas (there are uncountably many of these) or the free product of one counterexample for each quasi-identity (there are countably many of these).