Some functions who would seem to intersect it on top: $y=x^2$, $y=x^3$, $x=0$, $y=xa$ for a high enough $a$, $y=\tan x$, $y = \frac{1}{x}$
For instance, $y = x^2$:
Curves that do not fit: $y=0$, $y = \sqrt{x}$, $y = \ln x$, $y = \cos x$, $y = \sin x$.
For instance, $y = \sqrt{x}$
Naturally, this depends on the extent to which one zooms in or out on the origin. But I'd like to know how, for instance, we could speak of an infinite outwards zoom, or an infinite inwards zoom on the origin. Some graphs look exactly the same whether we zoom in or out! $x = y$ for instance.
A close enough zoom on $x^2$ makes it such that it stops intersecting the square on top:
An excellent example given by user Mason in the comments is: https://www.desmos.com/calculator/y3zeqnnybk What functions would intersect the top side on this square as it gets infinitely bigger?
So the top of your square is $y=k$ and what we want is that there exists some constant $C$ such that $\forall k>C, \exists x<k$ such that $f(x)=k$. I'm not sure that this characterization is any more than interpreting your geometrically posed question into symbols.