I would call it a linking verb. "Taller" is a predicate adjective that links back to the subject "he".
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'He will grow taller.'
Is the verb 'grow' used as a linking verb or an intransitive verb?
The verb is "will grow". The word "taller" describes how the growth will take place.
Does that help?
In this context, I do. But it's not worth arguing about. I will defer to the grammar experts.
You can remove taller and still have a complete sentence.