More thoughts on if a entire function has at least one coefficient of the Taylor series being real

Question

I have asked a question few days ago:

Entire function where at each $a \in \Bbb C$, at least one coefficient of the Taylor series at $a$ is real

The question is:

Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be a holomorphic function. Suppose that for each $a\in \mathbb{C}$ at least one coefficient of the Taylor's series $f$ about $a$ is real. Then, prove $f$ being a polynomial?

Thanks for @MartinR and @Conrad for their excellent explanation.

Today, I was reviewing the proofs. I recall that the Liouville's Theorem stated that is $f$ is a bounded entire function, then $f$ is a constant.

And because being a polynomial implies that there is some $N$ such that for all $k>N$, the derivatives are $f^{(k)}(z)=0$.

I am wondering if there is a way to prove the question with using the Liouville's Theorem.

Does anyone can help me figure it out?