The pile of garbage had grown (to) half his height.
To me, "had grown half his height" sounds natural. However, a native speaker told me I should add to.
Should I add a to? Why or why not?
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Yes, it needs "to". Something can grow to a measure, or it can grow by a measure.
"Grown to half his height" means that, as it stands now, it is half his height.
If it said "grown by half his height" that would mean that its existing height had increased by whatever half his height was.
Without a preposition, "grown" would mean that the subject had grown something else, for example, "he had grown some sunflowers".