Recall that a chain complex $(C_*,d)$ of abelian groups is contractible if it is homotopic to the zero map. Or equivalently: there exists a degree 1 map $F: C_* \to C_*$ such that $\operatorname{Id}= dF+ Fd$.

Question: does there exist a topological space $X$ which is not contractible (in the sense of topology), but whose complex of singular chain $C_*(X)$ is contractible?

More generally, one can ask whether the functor $X \mapsto C_*(X)$ from the category of topological spaces to the homotopy category of chain complexes of abelian groups forgets any information. I assume the answer is "yes", but I can't seem to come up with a counterexample.