We know that a space $X$ is contractible iff for any space $Y$ and for any two maps (continuous) $f,g:Y \to X$ we have that $f$ and $g$ are homotopic. We also know that the unit circle $S^1$ is not contractible so there must be a space $Y$ and two non homotopic maps $f$ and $g$ from $Y$ to $S^1$. I know it seems simple but I am new to the subject and I could not come up with an example of $Y$,$f$ and $g$ as above!
(note that I am talking about homotopy of maps not homotpy of paths)
Just take $Y=S^1$, $f=\operatorname{id}$ and $g=\operatorname{const}_p$, where the latter denotes the constant map with value an arbitrary point $p\in S^1$.
By the way: These are paths in $S^1$, but paths are functions so if you want some example of non-homotopic functions you are perfectly allowed to take non-homotopic paths.