I have two normal variables $X,Y$.
Is it possible to have $X = f(Y,Z)$ for some non-trivial function $f$, an independent normal variable $Z$ and that $X$ will be independent of $Y$?
You're not being that precise about what's independent of what. However, let's assume you mean the following: $Y$ and $Z$ are independent random variables and are normally distributed. Is there a non-trivial function $f$ such that $X=f(Y,Z)$ is both normally distributed and independent of $Y$? The answer is yes. Suppose $Z$ has mean $E[Z]=0$. Then any function $f(y,z)=g(y)z$, where $g(y)=\pm 1$ everywhere, meets your condition: regardless of the value of $Y$, $X$ has the same (normal) distribution.