Is there a function that looks like this? The idea is to find a family of functions where infinitely many of them converge to a point (pictured in orange) and the curves eventually have more distance as between one another as you travel away from the point. However, none of the curves intersect.
If the curves are differentiable in the orange region, it would be great, but if not, piecewise functions will also work. Ideally, a single $f(x,y)$ such that each curve is $f(x,y) = c$ for some $c \in \mathbb{R}$ is the most desirable.
There could be many examples consisting of the level curves of some function $f(x,y)=c$.
For example, for $f(x,y)=\left\vert\dfrac{1}{(x-1)(y-2)}\right\vert$ the level curves aproach the point $(1,2)$ in the manner you describe for $c_{2k-1}=2^k$, $c_{2k}=2^{-k}$ when $k$ is a positive integer. In fact, they approach the graph of $(x-1)(y-2)=0$.