Let's say you're in an interview and the interviewer leans forward and says:
"I want to get to know you better."
In this context, which is the verb? My initial reaction is: Want - auxiliary verb To get - main verb To know - modal verb
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A sentence requires a subject and a predicate. A predicate is a verb plus whatever other elements make up a verb phrase. In this case, the subject, clearly, is I, while the predicate is the verb describing the state or action of the subject. In this case, that is want, in a transitive capacity. Everything after that is the direct object of want, a noun in effect, which can be replaced by a pronoun. In other words, this sentence can be reduced to I want it. The verb phrase itself contains two additional verb phrases, but the verb that is serving the subject is want.
I want to get to know you better.
There are three non-auxiliary verbs, so there are three clauses.
As for the rest,
EDIT: All clauses have predicates, and in these clauses the verbs are the predicates.
Predicate is a logical term; logically, the predication structure of the sentence above
(without "better", which is semantically complex, ambiguous, and irrelevant) looks like this:
No, there is no auxiliary or modal verb; they are all lexical verbs. "Want" is the main clause verb. The other two verbs, "get" and "know", head subordinate non-finite clauses embedded within the main clause.
I want [ to get [to know you better]].
I would treat your example as a catenative construction. The term ‘catenative’ is derived from the Latin word for chain, for the construction is repeatable in a way that enables us to form a chain of verbs in which all except the last have a non-finite complement:
Your example then comprises a main clause (the whole sentence) which contains two subordinate non-finite clauses (bracketed) functioning as catenative complements:
to get to know you better = catenative complement of “want”.
to know you better = catenative complement of “get”.
The only direct object is “you” - it's the object of “know”, and the adverb “better” is a degree adjunct modifying "know”.