1. At the age of nine years old, he became too sick.

2. At the age of nine years, he became too sick.

3. At the age of nine, he became too sick.

4. At nine, he became too sick.

5. At nine years old, he became too sick.

6. At nine years old, he became too sick.

7. When he was nine years old, he became too sick.

8. Being nine years old, he became too sick.

9. Nine years old, he became too sick.
================
Thank you for your help. I have posted several sentences. Which ones are not acceptable in terms of grammar?

4 is not acceptable. At nine WHAT? Nine o'clock?

8 is not, either. It makes it sound as if being nine years old is the cause of the illness.
9 is grammatically correct, but rather imprecise. The meaning is clear, but one must assume some missing words. "At nine years old, he became too sick" is better. "He was nine years old when he became too sick" is even better.

All of the sentences you have posted are grammatically acceptable. However, the phrasing and wording vary slightly between them. Let me break it down:

1. At the age of nine years old, he became too sick. (repetition of "years old" is not necessary but still grammatically acceptable)
2. At the age of nine years, he became too sick. (repetition of "years" is not necessary but still grammatically acceptable)
3. At the age of nine, he became too sick. (simple and concise)
4. At nine, he became too sick. (simple and concise)
5. At nine years old, he became too sick. (clear and straightforward)
6. At nine years old, he became too sick. (same as sentence 5, just presented twice)
7. When he was nine years old, he became too sick. (clear and establishes a time frame)
8. Being nine years old, he became too sick. (phrased in a different way, but still correct)
9. Nine years old, he became too sick. (phrased in a different way, but still correct)

In terms of grammar, all of these sentences are acceptable, but the choice of phrasing may depend on the context or style preference.