Hi. It's an interesting problem and perhaps someone here can suggest a more elegant and non-overkill way to prove this than what I suggest here:
I assume you mean "entire" function which can either be a polynomial or a non-polynomial. If it's a polynomial, then by the Fundamental Theorem of Algebra, the polynomial reaches all values in the complex plane exactly n times where n is the degree. If it's not a polynomial, then by Picard's First Theorem it has an essential singularity at infinity and by Casorati-Weierstrass, in any neighborhood of that singularity, the function comes arbitrarily close to any number in C, that is, it's dense in C.