Is there some example of that the dimension of every fiber bigger than the relative dimension?
Solution 1:
In part c of this exercise, i.e. II.3.22(c) in Hartshorne Algebraic Geometry, the reader is asked to show that there exists a dense open subset $U$ of $X$ such that for any $y\in f(U)$ we have $\dim U_y=e$. Since $U$ is in particular nonempty, this means that no example exists where the dimensions of all of the fibers are $>e$.
Note that the argument works without the hypothesis that $X$ and $Y$ are of finite-type over a field $k$. We just need $f$ to be of finite-type, and we take $e$ to be the transcendence degree of $K(X)$ over $K(Y)$.