Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?
Yes. In fact, there exists such an $f$ taking every value uncountably many times. Take a continuous surjection $g: [0, 1] \to [0, 1]^2$. (Such things exist: they're spacefilling curves.) Then the composite $f$ of $g$ with first projection $[0, 1]^2 \to [0, 1]$ has the required property. 


Take a projection of a wellknown Peano curve, which is a surjective continuous mapping $[0,1] \to [0,1]^2$ to a factor. 

