I believe the only addition to the excellent answer by @AndrewLeach's answer is [God] cannot deny himself [e.g. sin, break his promises or oaths], the thing we do when we sin. According to Scripture and therefore Catholic Teaching, this appears to be the ONLY thing that God CANNOT do.
What can God really NOT do?
"Nonsense remains nonsense even if we talk it about God." C.S. Lewis:
The Problem of Pain | Cf. This AndreasBlass' answer.
Is there really a limit to God's power? What does the Bible say? What does the Church teach?
- "For with God nothing will be impossible.”
- "Is anything too hard[wonderful] for the LORD?"
- But thou art merciful to all, for thou canst do all things[.]
- Our God is in the heavens; he does whatever he pleases.
- [If] we are faithless, he remains faithful — for he cannot deny himself.
To me, according to the Bible, and therefore what the Church must teach [CCC 268-278], nothing is impossible for God and the only thing he cannot do, is deny himself.
Whence then comes Nonsense remains nonsense even if we talk it about God (cannot make a square/circle, make a stone too big for him to carry, etc.)?
From Christian reasoning, thinking, theology, and philosophy.
On the contrary, It is said: "No word shall be impossible with God" (Luke 1:37).
I answer that, All confess that God is omnipotent; but it seems difficult to explain in what His omnipotence precisely consists: for
there may be doubt as to the precise meaning of the word 'all' when we
say that God can do all things. If, however, we consider the matter
aright, since power is said in reference to possible things, this
phrase, "God can do all things," is rightly understood to mean that
God can do all things that are possible; and for this reason He is
said to be omnipotent. Now according to the Philosopher (Metaph. v,
17), a thing is said to be possible in two ways. - Summa Theologica >
First Part > Question 25 > 3. Is He omnipotent?
I have of course a tremendous admiration and respect for the Angelic Doctor, and most likely I have failed to grasp his argument. I only note here that his theological thinking has not made it to official Church teaching.
Can God therefore make a square/circle, make a stone so big he cannot lift?
The answer lies in AND. Before the incarnation and even after, God becoming man - “We stone you for no good work but for blasphemy; because you, being a man, make yourself God.” - and to die on the cross, is a stumbling block to Jews and folly to Gentiles.
Therefore, for the same God, as man, Simon of Cyre′ne was compelled to carry his cross.
What about square/circle?
These shapes are mathematical concepts developed to help in understanding physical reality. A string in the shape of a circle and the same string turned into a square shape, I would say easy for God.
What has led to this thinking?
In physics: the wave-particle duality of light in particular and of matter in general.
In Biology: egg-laying mammal and innumerable/countess examples in the animal kingdom like parthenogenesis, sex in crocodiles determined by the nest temperature of the eggs, etc.
Mysteries of Faith: Jesus: God-made-man; Jesus' Substantial presence in the Eucharist, the elect get to see God, etc.
Church treasures and artifacts: The Shroud of Turin; The image of Our Lady of Guadalupe.
For with God nothing will be impossible. He remains faithful — for he cannot deny himself. - I personally would be extremely careful to even think that there is even a tiny bit limit to God's unbounded/infinite power.
I see no where in Scripture or Church teaching Nonsense remains nonsense even if we talk it about God.