Truth be told, I never graduated from college, and this is the closest I've ever gotten to a college graduation.
Almost, but not quite. This sentence does indeed involve a relative clause. We could reconstruct the relative clause like this:
- Truth be told, I never graduated from college, and this is the closest that I've ever gotten to a college graduation.
The important thing about relative clauses is that they have a gap in them. The gap tells us where the missing element should be. So for example consider this sentence:
- This is the man that [you saw yesterday].
This relative clause has a gap in it that tells you where the man would be in the clause:
- This is the man that [you saw ___ yesterday].
Because our language brain knows where the gap is we can understand the sentence like this:
- This is the man that [you saw the man yesterday].
The relative clause
It is not straightforward to understand exactly where the gap is here. This is because, as the Original Poster felt, we often associate relative clauses with a noun phrase antecedent. However, in this case the antecedent isn't a typical noun phrase. Here's where the gap is:
- this is the closest that [I've ever gotten ____ to a college graduation]
We understand it like this:
- this is the closest that [I've ever gotten close to a college graduation].
This sentence is unusual because it has a superlative adjective as an antecedent, not a noun or clause. The verb gotten her means something like come or been.
The grammatical function of closest
The Original Poster asks if the closest is the Object of the verb GET, in this case part of the perfect construction have gotten. The answer is, no. However, it has a function similar to an Object. Like an Object, it is a Complement of the verb. But it is a Predicative Complement, not an Object. Like Objects, Predicative Complements fill a special slot set up by the verb. They don't tell us what someone was doing something to though and they don't introduce a new entity into the sentence. Instead Predicative Complements tell us something about the Subject or Object of the sentence.
In this case, the phrase the closest that I've ever got to a college graduation is the Predicative Complement of the verb is. It describes the situation denoted by this. Notice that this is NOT an adverbial. It is not an extra part of the grammar in the sentence. It is not an extraneous addition to the meaning either. The sentence is quite simply ungrammatical without it:
*Truth be told, I never graduated from college, and this is. (ungrammatical with this meaning)
The verb BE cannot ever, ever take Objects, it can only take Predicative or Locative Complements.