We would need to clarify what you mean by asking for a counterexample. There is no point in looking for an object that is not identical with itself unless you start with some criteria of what identity is.
In logic, identity (or equality) is a dyadic predicate. We could write it as Identical(x,y) but it is more handy to use some syntactic sugar and write it as x=y. We normally have to 'interpret' predicates, along with names and functions. Interpretation here is a technical term from model theory. It is a function that, for a given domain, assigns referents to names, functions to function-symbols, and relations to predicate-symbols.
Now we have a choice. We could treat identity like any other predicate and interpret it. This is called "first-order logic without identity". On this approach, it is down to the theory to provide the axioms that will determine whether x=y is true for some x and y. Such axioms typically have to do with reflexivity, symmetry, transitivity and substitutability. They are spelled out in a little more detail in this article.
But with the vast majority of theories, we end up using the same axioms for identity, so we can help ourselves to a shortcut and treat identity as a logical constant. This is called "first-order logic with identity". It transfers responsibility for defining the properties of the identity predicate from the theory to the logic. Under this convention, the law of identity is a logical truth.
But occasionally, just occasionally, we may come across a domain of discourse where our requirements concerning identity are different from usual. We may instead wish to interpret identity using some other equivalence relation. So we can stick with first-order logic without identity and interpret the identity predicate. Under some such interpretations we may find cases where distinct individuals a and b satisfy a=b. An interpretation where this happens is said to be non-normal, while an intepretation where it does not happen is said to be normal.
So, your question, Is there a counterexample? really amounts to asking, is there some domain of discourse in which it makes sense to use an interpretation of identity with non-normal models? The short answer is yes. The longer answer involves philosophical arguments about the nature of identity.
In quantified modal logic, it may be that an individual exists in more than one possible world, and we require an identity predicate that operates cross-world, and another that operates intra-world. When dealing with intensional contexts, we might require an identity predicate that is substitutable in such a context, and another that is not. As has been mentioned in the comments, in the case of the elementary particles of quantum mechanics, it is arguably impossible to speak of the identity of individual particles, since they are indistinguishable, even in principle. This is a motivation for Schrödinger logic. There is a longer discussion of this in this Stanford Encyclopedia article.
One last point. Rationalism is BS. You cannot make progress in logic, or any other discipline, by just saying, "Well this is just obvious to me, so it must be so." Our knowledge, including our knowledge of logic, is the product of centuries of blood, sweat and tears by lots of clever people. Nothing is obvious. Everything has to answer to critical enquiry. The history of philosophy is littered with things people used to think were obviously true, but which to us now would seem not just false but absurd.