Let $k$ be an algebraically closed field and $\mathbb A^2_k=\operatorname {Spec}k[x,y]$ the affine plane over $k$.
Consider the ring $R \subset k(x,y)$ of the rational functions on the plane defined and constant on $V(x)$ (the $y$-axis $x=0$).
What is $\operatorname {Spec}R$ ?
(This is the geometric translation of an example due, I think, to Krull for which I unfortunately have no reference.)

Edit
Sorry,my definition of $R$ above is a bit ambiguous. What I mean is that $R$ consists of those fractions $r(x,y)=\frac {p(x,y)}{q(x,y)}$ which can be written as the quotient of two polynomials $p(x,y),q(x,y)\in k[x,y]$ such that $q(0,y)\neq 0\in k[y]$ and $\frac {p(0,y)}{q(0,y)}\in k\subset k[y]$.
For example the rational function $\frac {y+x}{y-x}$ mentioned by @YCor in the comments does belong to $R$ since $y-0\neq0\in k[y]$ and $\frac {y+0}{y-0}=1\in k \subset k[y]$