> *We had* [*to go* [*to get a book*]].

Preliminary point: most speakers treat stative "have" as a lexical verb, but some treat it as an auxiliary.

The sentence consists of a main clause (the sentence as a whole) and two embedded subordinate clauses, as bracketed.

"Have" is a catenative verb and the subordinate clause "to go to get a book" is its complement. 

The second subordinate clause, "to get a book", is a purpose adjunct in clause structure. "A book" is object of "get".