Given a rectilinear polygon, what is an run-time efficient algorithm that finds the largest inscribed rectangle the sides of which either parallel or perpendicular to the sides of the rectilinear polygon? A dumb algorithm to form all the rectangles formed from unordered 4-tuples of the edges, for each of the rectangle check to see if any edge or vertex lies inside it, then choose the one with the largest area. The run time is quintic.
Daniels, Milenkovic and Roth  show how this can be done in $O(n\log^2 n)$ time ($n$ being the number of vertices) even for general polygons (possibly with holes). They also mention that the algorithm from Aggarwal and Suri  for largest empty rectangle can be adapted to your problem in the rectilinear case, but I haven't thought about how one would do that.
 Daniels, Milenkovic and Roth. Finding the largest area axis-parallel rectangle in a polygon. https://www.sciencedirect.com/science/article/pii/0925772195000410
 Aggarwal and Suri. Fast algorithms for computing the largest empty rectangle. https://dl.acm.org/doi/10.1145/41958.41988
After seeing @Tassle's answer, I did a bit Google search and found that the largest empty rectangle is a well researched subject.
In the following, I try to draft an algorithm on my own.
Suppose the polygon has $2n$ sides. There has to be $n$ horizontal and $n$ vertical sides. For simplicity sake, we assume no two horizontal sides of the rectilinear polygon have the same $y$ coordinates, and the same for the vertical sides. A candidate rectangle has to have all its sides coincide with the sides of the rectilinear polygon. Given the top and bottom sides of the candidate rectangle, the left and right sides are determined or the candidate rectangle does not exist. So we need to examine at most $\displaystyle n\choose 2$ candidate rectangles. The complexity of this algorithm is at least quadratic.