I got him to repair the bike.
In this sentence is the infinitive functioning as a noun or an adjective?
Or without context is it just an <object + object compliment>?
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I got him [to repair the bike].
This is called a 'catenative construction'.
The verb "got" is here a catenative verb and the bracketed subordinate infinitival clause is its catenative complement.
The intervening noun phrase, "him", is the syntactic object of "got" and the understood (semantic) subject of the subordinate clause.
Incidentally, infinitival clauses do not function as nouns or adjectives, but as subjects, adjuncts, complements (as in your example) or as modifiers. The classification of subordinate clauses is based on a number of factor such as their internal form, not on spurious analogies with the parts of speech.
It is a complement, (note spelling with -e-) that is a construction that completes the clause. It is not a noun nor functioning as one. The di-transitive "get" with two objects has a different sense of "acquire", whereas this means "cause". And it isn't an adjective nor functioning as one, as it doesn't describe a noun (Neither "I" nor "him" could be described as a "to-repair-the-bike person)
The verb "get", in the sense of "cause" has a structure of "get (object) (to infinitive)".