Infinite entropy

Question

I am learning about entropy, and am slightly confused by an exercise in a book. It asks if there exists a discrete random variable $X$ with distribution such that $H(X)=\infty$. My intuition is that there is not, as the set is always countable as it is a discrete random variable, and the probability is less than one, and the definition for entropy is $$H(X)=-\sum_{x\in{X}}P(X=x)log(P(X=x))$$ Any help understanding this would be appreciated. Thank you.