Infinity-to-one function
Consider a space filling curve $\gamma: I \rightarrow I^2$, the projection $q: I^2 \rightarrow S^2$ given by the quotient topology on the square that furnishes the sphere, and the projection $\pi: I^2 \rightarrow I$ on the first coordinate.
The map $q \circ \gamma \circ \pi \circ \gamma$ satisfies what you want.