As an example in The Emperor's New Mind he makes a number of claims about consequences of Gödel's theorem for human thought that were flat out wrong. I remember reading criticism of it from a logician in the mid-90s, and Penrose's response. While I'm no logician, I know enough about it to know that the logician was right and Penrose was wrong. Human thinking is known to accept inconsistencies and so you can't apply Gödel's theorem to derive any interesting conclusions.
When he moves from logic to physics he is even more out of his depth. Given his verifiable mistakes in logic, I am comfortable in concluding that his startling assertions about physics are far more likely to arise from further mistakes of his than from his having found serious reasons to question the foundations of modern physics.