I was pondering this question while writing on whether or not God had the ability to create a best of all possible worlds. I hold that God is not limited by anything (a view among classical theists such as myself). But I also hold that God can't do things such as make a rock so heavy he can't lift it, or he can't make a married bachelor, due to the laws of logic, such as the law of non-contradiction. So is God limited by these laws of logic?
Many, if not most classical theists think so. Aquinas solved a lot of "problems" by citing God's logical nature as being the ontological source of things like the law of non-contradiction. God not being limited by anything is a somewhat separate issue, but many classical theists (for example, William Lane Craig) would argue that God's limitations are not limitations in the normal sense of the word (as in, he's limited because he can't make a rock he can't lift) but rather logically incoherent statements that can't be spoken about as actions at all. Another way a classical theist might think about it is that God cannot act against his nature (law of identity), and because God's nature is to a classical theist the ultimate reality, God acting against his nature is both a logical impossibility and an incoherent statement.
Our ideas of logic is limited to what we have experienced. Some things might seem like it would defy logic, but we haven't experienced enough knowledge to understand an answer. For example, many people thought the theory of relativity defied the laws of logic (and physics) in many ways. But there was a way around it using ways not yet thouht about.
Also, if you believe in God, you must believe in many things about him that seem illogical.
The Wikipedia article on omnipotence explains most of the subtleties, writing classical theists: https://en.m.wikipedia.org/wiki/Omnipotence
In response to questions of a deity performing impossibilities, e.g. making square circles, Aquinas says that "everything that does not imply a contradiction in terms, is numbered amongst those possible things, in respect of which God is called omnipotent: whereas whatever implies contradiction does not come within the scope of divine omnipotence, because it cannot have the aspect of possibility. Hence it is better to say that such things cannot be done, than that God cannot do them.
The possible paradoxes are also well explained in Wikipedia: https://en.m.wikipedia.org/wiki/Omnipotence_paradox
The paradox arises, for example, if one assumes that an omnipotent being has no limits and is capable of realizing any outcome, even a logically contradictory one such as creating a square circle
It is also well hashed through in prior questions of this forum, such as Can God make a rock so heavy he cannot lift it?, Problems with the Omnipotence paradox, Does this explain the Omnipotence paradox?, Can someone explain omnipotence to me? ... It does not seem the current question brings anything new to the table that a quick Google search could not solve.