Can topological entropy be infinte? I wonder if the topological entropy as defined by Adler or Bowen can be infinity.
Can you answer that?

This question is actually answered in an example of Adler's original paper. Let $X$ be any compact Hausdorff space with infinitely many points, let $X^\infty$ be the space of two-sided infinite sequences $(x_n)_{n=-\infty}^\infty$ with $x_n \in X$, equipped with the product topology. Then the shift $\sigma:X^\infty \to X^\infty$, $\sigma((x_n)) = (x_{n+1})$ has infinite entropy.
