I learned about this from Matt Parker's Stand-Up Maths channel. It was originally conceived as a counterexample, a sorting algorithm that was obviously broken, but it does actually sort correctly. The algorithm:

for i = 1 to n do  
	for j = 1 to n do  
		if A[i] < A[j] then  
			swap A[i] and A[j]  

It has a few quirks (like j accessing elements outside of i's range, and the A[i] < A[j] comparator being backward) that should break it, but they all work together to make the algorithm correctly (if inefficiently) sort the input.

paper describing the algorithm in more detail.