Grammaticality of "What is there there?"
If someone says
I am going to the market
I may ask
What is there at the market?
If someone says
I am going to the bookstore
I may ask
What is there at the bookstore?
If someone says
I am going there
is it then correct to say
What is there there?
It seems correct following the pattern, but it sounds weird.
Solution 1:
That sentence is perfectly fine. The two "there"s here are different types of word. The first is a type of Noun Phrase. The second is a Locative Preposition (some people think it's a Locative Adverb). It indicates a location.
The first there is the Subject of the question. The second is a locative Adjunct ( - where Subject and Adjunct are different functions in the sentence).
With questions like this, the Wh- word moves to the front of the sentence. If the Wh- word is not the Subject of the sentence, then the Subject of the sentence inverts with the Auxiliary Verb. In this case the auxiliary is the Verb BE. For all questions like this, there is a non-inverted form - consider it an ugly sister - where the Wh- word stays in the same position it would be in in a normal sentence. It's sometimes called an in-situ question. The in-situ question for the Original Poster's example is:
- There is what there?
Here we can clearly see the Noun Phrase There sitting in the Subject position in the sentence. We call it a "Dummy" Subject here because it has no lexical meaning. The Verb is is where we expect to see it, right after the Subject. What, the internal Complement of is, appears in its normal position after the Verb. Lastly the Locative Adjunct there takes up its usual position at the end of the sentence, where we expect Adjuncts to be.
This all goes to show that the Original Poster's example is well formed, and has all the correct parts that we should expect it to have.
Solution 2:
"What's there there?" is grammatical.
Here's why.
"There is a book." It means the existence of a book in a certain place, which is here not stated.
"There is a book there." This means, "You got a book there (here, a place)."
So the first "there" in your example is just about existence; the second the place.