As a native speaker, I had no idea what was being said. Seems like a very extreme example. Similar to the shi shi poem in chinese.

The "had had" example is much more relevant to someone learning English. Often when I construct a sentence with "had had", I question whether it looks right and if I can rephrase it.

As a native speaker and an obsessive reader, no, I had no clue. I can work it out now that I've read the description, but it's definitely not natural.

They wouldn't understand the sentence when reading it, but may when listening to it spoken naturally, with correct tones, stresses and pauses, if they're already familiar with all three uses of the word "buffalo". Not many people know of the city Buffalo, or use the verb "to buffalo" so that one's iffy. Most people speaking it would use the word "which" to introduce the restrictive relative clause to make themselves better understood. To ensure native English speakers not familiar with all three uses of "buffalo" understand it when reading it, many people would write it "Buffalo-buffalo, which Buffalo-buffalo buffalo, buffalo Buffalo-buffalo."

The 'had had' versions are actually a feasible examples, and don't rely on cutting out other words to make them work. Still, it's uncommon to put so many had's together.

You probably wouldn't use four had's in that particular example though - "All the faith that he had, had had no effect..." or "All the faith that he had had, had no effect...".