When LRU is NOT the optimal replacement policy
Consider a program that plays a video in an endless loop (e.g. an electronic advertising billboard), assuming the RAM is constrained enough that the whole video can't fit at once.

The most recently used page of memory will contain the most recent section of the video, that just played. LRU would be the worst choice, because you're holding onto the parts that have already played, which are the parts that won't be needed for the longest time. They won't be needed again until all other pages are played first, and the video repeats. By definition, such a use case is hitting the least recently used pages, over and over.

The optimal page replacement policy in such a situation is one that either preempts the future pages (which is possible in this case because of the periodicity and simplicity of this usage profile) or one which permanently  holds onto some parts of the video (doesn't matter which parts, as they're all played equally) and uses the rest of the space to cycle through the rest of the pages.