I don't know how you want to define "pathological", but some of the corollaries of Runge's theorem give you functions with interesting properties. One of mine: there is a rational function $f$ such that for every holomorphic function $g$ on the open unit disk $\mathbb D$, $g$ or $g-f$ has a zero in $\mathbb D$. This is American Mathematical Monthly problem 6520, solution at www.jstor.org/stable/2323638