Example in which light takes the path of maximum optical length According to the modern version of Fermat's principle,"A light ray in going from point A to point B must traverse an optical path length that is stationary with respect to variations of that path.".Is a maximum optical path length possible ?What if we keep adding deviations to the optical path length?
 A: I offer you two examples.
A) Draw a semicircle on AB as its diameter. Assume semicircle is
reflecting, and you are looking for reflection from A to B. It obviously
happens at C, midpoint of arc AB. I leave to you to show that for any other point P of arc ACB you have AP + PB < AC + CB.
B) Consider two points A, B, with A in vacuum, B in a medium with
refracting index $n$. The locus of points P such that ${\rm
AP}+n\,\rm BP=const$ is a curve (Cartesian oval). If the oval separates vacuum and medium with index $n$, then all rays APB have the same optical length. Let C be the intersection of the oval with AB. Draw a circle having center on AB, between B and C. If medium is delimited by this circle, then for all P on the circle ${\rm AP}+n\,\rm BP < AB$.
A: You are quoting wikipedia
There is no maximum length from point A to point B (the path could be arbitrarily long), the more deviation to that minimum path the more length is added in such a way that the light phase is so mixed up that it ends up cancelling itself (no light).
EDIT: Quantum electrodynamics from Feynman offers a nice accessible explanation for this phenomenon
