You have a compound noun "fish [that] fish eat". Call that X. Then you have a larger compound noun, "fish [that] X eat". And as with the original you can tack an 'eat' on the end as those fish eat some unspecified food. Of course you can then go further and define Y as "fish that X eat", etc. It can be infinitely recursed.
It's convenient using "X" as a placeholder, because that specifically masks the problem of stacking up verbs.
"Fish fish eat" is indeed a sensible noun phrase. And we can make that noun a subject of the verb eat. But "fish fish eat eat" is not a noun phrase, and we can't make it the subject nor the object of a further "eat".
"fish fish eat eat" isn't a noun phrase, no. The noun phrase in question is X = "fish fish eat". Which you can embed between "fish" and "eat" to make a bigger noun phrase:
(1): X' = fish X eat ("fish that X eats")
You can continue forever by repeatedly applying equation (1) to make bigger and bigger noun phrases. Producing:
(2): X = fish fish eat
(3): X' = fish {fish fish eat} eat
(4): X'' = fish {fish {fish fish eat} eat} eat
At the end of which you can stick an "eat" on the end to make a sentence.
First of all, that wasn't the original sentence. What you've got is "fish fish fish eat eat", whereas the original was "fish fish fish eat eat eat".
It would be nice if English worked recursively like that, but it doesn't. Maybe I was wrong in focusing on the verbs; the problem is also stacking nouns the way the sentence does. By splitting the object, subject, and verb so dramatically, the sentence doesn't end up making sense.
Obviously if you take it as axiomatic that this algorithm does work, then you can keep claiming it's grammatical. But even the first round (fish fish fish eat eat) doesn't add up. The word "that" isn't a mere helper; there is no English sentence in which "fish fish fish" (all as nouns) can make sense.
The sentence is very difficult to parse, yes, but it is grammatically correct. You can make a noun phrase with an arbitrary number of repetitions of 'fish', call that F, added to F-1 repetitions of 'eat'. IE, 'fish fish fish eat eat', or 'fish fish fish fish eat eat eat'. Each of those is one big noun. You can make it as long as you want.
I'm not sure how to demonstrate this any more clearly than it already has been, but I'll try one last time. "fish that fish eat" is a compound noun. I think we agree on that. And you can shorten that to "fish fish eat", and it's still a grammatically correct noun. "fish fish eat" is a thing. Now you can build a sentence around that thing. For example, "The water fish fish eat swim in is cold." Right? It's easiest to parse if you say the 'fish fish eat' noun more quickly than the rest of the sentence, like, "The water fish-fish-eat swim in is cold."
Of course, we can replace the noun water there with more fish! "The fish fish fish eat swim with are small." In fact, you don't need the 'the' there. "Fish fish fish eat swim with are small," works too. As does, "Fish fish fish eat eat are also small." Note we just replaced 'swim with' with 'eat' there. Now we have actually built a larger compound noun, "Fish fish fish eat eat." Finally, we can replace the verb 'are' at the end and we get, "Fish fish fish eat eat eat," a grammatically correct sentence.
Fish [that] fish [that] fish eat eat... eat.
You have a compound noun "fish [that] fish eat". Call that X. Then you have a larger compound noun, "fish [that] X eat". And as with the original you can tack an 'eat' on the end as those fish eat some unspecified food. Of course you can then go further and define Y as "fish that X eat", etc. It can be infinitely recursed.