Complement of the object?
Solution 1:
Some terminology in this area is confusing or varies between authors, so I’ll start with at attempt at the definitions that I use.
Object: a noun phrase that acts as a specific kind of complement. All objects are complements and noun phrases, but I think these criteria might not be sufficient to establish that something is an object.
Complement: something that completes the meaning of the head (or in your case, completes the meaning of the verb). A required element is always a complement, but complements aren’t always required to be overtly present. For a better definition, see here: https://linguistics.stackexchange.com/questions/20717/phrasal-verb-adjunct-vs-complement
Adverbial: an optional element of a clause or verb phrase that adds additional meaning rather than completing the existing meaning. Adverbials are also called “adjuncts”, which is a term that can also be applied to optional “adjectival” elements. There is no such thing as an “adverbial object” as far as I know. “Adverbial” is not the same thing as “adverb”: “adverb” is a class for words while “adverbial” is a function for phrases (adverbial phrases can be headed by words of various classes other than adverb).
As you said, the sentence makes sense without “100 times”. “100 times” does not complete the meaning of “punched (him)”, so it is not a complement. It is adverbial.
“Red” is a complement in the phrase “paint it red”. I think another example of a complement taken in addition to an object would be “down” in “punch him down”.
Solution 2:
I punched him 100 times.
"100 times" is neither an adverb phrase nor a complement. It's a noun phrase functioning as a bounding adjunct of frequency, an optional item in clause structure.
We know it's not a complement because it's not required to syntactically complete the verb phrase, and nor is it licensed by the head.
You also asked in comments about:
We watched him laughing.
"Laughing" is a subordinate non-finite clause functioning as complement of "watched" (not of "him"). Syntactically, "him" is object of "watched" and the semantic (understood) subject of the subordinate clause.
"Him" is called a 'raised' object because the verb that "him" relates to is higher in the constituent structure than the one it relates to semantically. Note that generally only NPs ("They elected him treasurer") and AdjPs ("I painted the house white") can be objective complements.