I think it should go along these lines: assume g is not a polynomial. then it has an essential singularity at infinity (by which i mean that g(1/z^\bar)^\bar has essential singularity at zero -- you can make this precise). by casorati-weierstrass its image of some neighborhood of infinity is almost the whole complex plane. now image of almost-the-whole complex plane under any entire f is either the whole complex plane or a point. it cant be just a point since f(g(x)) is not constant. but f(g(x)) is a polynomial, so it has a pole at infinity, so image of a neighborhood of infinity must be a neighborhood of infinity.

sorry for sloppiness and vagueness. if u want i can try making this precise.