Is this an adjectival or adverbial phrase?

Question

I want to know everything there is to know about you.

Does ‘to know about you’ modify everything (adjectival function) or is it an adverbial phrase? Or are both possible?

Answer 1

There are three interlocking pieces here.

One of them is a ‘fossil’ construction which has hung around for centuries: a marked infinitive (to VERB) employed as a modifier signifying that the noun it modifies can be or should be VERBed. As a direct attributive it is placed after the noun:

a job to do
a race to run
a mountain to climb

Thus, although it employs the active infinitive, its sense is passive: a job to be done, a race to be run, a mountain to be climbed (and in fact that passive construction is gradually replacing the active one).

That is the sense of the second to know in your example: everything there is to be known.

The second piece is the there is. In the chunk of your example everything there is to know, the sub-chunk headed by there clearly means ‘everything which can be known’. What’s happening here is that there is to know is a relative clause (modifying everything) which is headed by the ‘null relativizer’ (sometimes written * Ø *)—a very common ‘absent’ relative pronoun replaceable by which or that:

everything that there is to know
everything which there is to know
everything Ø there is to know

The final piece is the participial phrase about you. I suspect you are thinking of the phrasal verb know about and the reduced clause I want to know about you when you conjecture that this phrase might be an adverbial adjunct. In fact, however, it is adjectival, modifying everything, as may be seen if we move the pieces around:

I want to know everything about you.
I want to know everything about you there is to know.
I want to know about everything there is to know … Oops! where are we going to put the you?

So what you have in your bolded chunk is one prepositional phrase modifying everything and the back half of a relative clause also modifying everything.