I have grown very fond of Alec.
I think this sentence looks weird.
I suppose grown is a verb, and should be followed by an adverb. But I don't think fond is an adverb.
Is this sentence correct in grammar? Thanks.
It is grammatically correct. From Longman Dictionary, "Over the years we’ve grown very fond of each other". In this sentence structure, the word "Fond" is an adjective. Grown acts like be. I have grown (been) fond of Alec. Gradual connotes a gradual process of fondness.
You can use the word grow plus an adjective, which means become. I have grown (increasingly) annoyed/satisfied. The sound was growing louder.
"to grow" is a copula verb or linking verb, which are connected with predicative adjectives.Google for a list of copula verbs, eg
The copula verbs are mainly to be, to become + variants and to stay + variants. A second group can be labelled verbs of perception (to look, to sound, to smell, to taste, to feel).
The rule that verbs are followed by adverbs is a beginners' rule and concerns only adverbs of manner. But the rule does not mention all the cases where you need an adjective after a verb.
MUCH more simply, 'very fond' is describing the the person (the noun: in this case "I"), not the action of becoming fond (the verb: "growing fond of") Therefore it is an adjective.
Compare: "I have quickly grown tired of confusing grammar rules" where quickly is an adverb because it relates to the action.
(i.e. the process of growing tired was quick, not "I" am quick)
Some verbs take Direct Objects:
In the sentence above, a carrot is the Direct Object. Now a Direct Object is a special type of Complement. A Complement is a phrase that fills a special slot (space) set up by the verb. There are other types of Complement that verbs can take. For example the verb GO takes Locative Complements. These tell us about where something is or where something is going:
Locative Complements are usually preposition phrases.
Another type of Complement is a Predicative Complement. A Predicative Complement gives us more information about the Subject or Object of the sentence.:
In the first sentence the Predicative Complement is happy. It describes the Subject of the sentence, Bob. In the second example the verb became is taking a Predicative Complement, angry. Again this describes the Subject of the sentence, the elephants. The last sentence is different. Here the verb made is taking two Complements. One is the Direct Object, the orangutan, the second Complement is the Predicative Complement happy. This time the Predicative Complement is describing the Direct Object, the orangutan, not the Subject, the flowers.
Predicative Complements are often adjective phrases, but they can also be noun phrases or preposition phrases:
Notice that in the first sentence there is only one person. The phrase a builder just describes the Subject, Bob.
However, Predicative Complements are almost never adverbs:
The Original Poster's sentence
I have grown very fond of Alec.
When the verb GROW is used with this meaning, it means something similar to BECOME. Like the verb BECOME, the verb GROW also takes Predicative Complements. These are nearly always adjective phrases (but they can be preposition phrases too).
In these sentences the adjective irritable is describing the Subject, Bob. Notice that we don't usually use adverbs to describe nouns. If we try to use an adverb as a Predicative Complement here, it will be ungrammatical:
In the Original Poster's example, the adjective phrase is very fond of Alec. It describes the Subject, I. We cannot use an adverb here, because this phrase is the Predicative Complement of the verb grew.