A noun phrase is only what you (or the linguist you learn from) define
it to be. Different understandings of a language could possibly lead
to different definitions of what constitute a noun phrase.
Given that you have some precise idea of what constitute a noun
phrase, you then have to express that idea so that you can communicate
it to others. To communicate it precisely, you need to express it with
some kind of formal (mathematical ?) structure or notation.
There are many ways to do that, a simple one being phrase structure
grammars introduced by Chomsky half a century ago.
Such a grammar is constituted of rules describing how phrases (a noun
phrase for example) are built by putting together (sub-)phrases and
parts of speech (words).
So for example you could add to your lists of examples
You should eat NP(NP, Prep, NP)
for the sentence
You should eat fish from the blue sea
This is an example of a noun phrase composed with two other noun
phrases "fish" and "the blue sea", and a preposition, according to a
recursive rule. Examples can be more complex and involve different
kinds of phrases.
Such recursive rules can allow you to make arbitrarily long noun
phrases, and thus you can have an infinite number of noun phrases,
though all defined with a finite number of rules (of ways to construct
them): I added only one rule to your set.
Now, whether you have an infinite or a finite number of phrases of a
given type depends on the way you understand your language and wish to
describe it. In general, people do understand languages as having an
infinite number of noun phrases, which can be characterized by a
finite number of rules.
I insist on the fact that it all depends on the linguist's
understanding because there is not "a complete list of rules or
something out there".
Actually there are usually many such lists for a given language, corresponding to different
understandings of the structure of the language, corresponding to
different linguistic theories. It does not show too much for simple
noun phrases, but the complete description of a language goes far
Furthermore, there are also different ways of expressing the structure
of the language, with rules or otherwise. The world is a complicated place.