Let $f$ be an entire function which takes every value no more than three times. What can it be? 
Let $f$ be an entire function which takes every value no more than
  three times. What can it be?

Consider the singularity at infinity. If it is removable then $f$ is constant. If it is a pole then $f$ is a polynomial, and it's clear that degree of $f$ is less than or equal to three. Suppose $f$ has an essential singularity at infinity. The Big Picard theorem says that it takes every value except for possibly one (call it $\omega$) in a neighborhood of $\infty$. Let $w\neq \omega$ and take some value $z_1$ where $f(z_1)=w$. Then there is a neighborhood of infinity which does not include $z_1$, and so there must exist another point $z_2$ where $|z_2|>|z_1|$ and $f(z_2)=2$. By induction there must exist a sequence $z_n$ of distinct points with strictly increasing moduli where $f(z_n)=w$, for all $w\neq \omega$. 
Edit: I suppose I should clarify why the degree of $f$ must be at most three. This should follow from the density of separable polynomials...suppose $f(z)$ is a polynomial of degree greater than three. If $f(z)$ is separable, then $f$ takes the value zero at $>3$ distinct points. If $f$ is not separable, then I want to say that there is a constant $b$ such that $f+b$ is separable, and then $f$ takes the value $-b$ at $>3$ distinct points. But I should make this rigorous...after all, separable polys may be dense, but how do we know that a set of inseparable polynomials is not parallel to the subspace spanned by the constant polynomials (so $f+b$ is inseparable for all $b\in \mathbb{C}$)? Any tips for this?
 A: First, did you mean there must exist another point $z_2$ ... such that $f(z_2)$ = w?  You said 2. 
It looks to me like the proof is correct; however I think it could be written more clearly:
You said if the singularity is a pole then f is a polynomial of degree 3.  Can you explain why it couldn't be a polynomial like p(z) = $(x-1)^3(x-2)?  Is 1 counted as 3 separate values?  Could you reference a theorem?  
You took a w for $z_1$; I would have found it easier to follow if you could stated more clearly why w can't be omitted from any closer neighborhood of $\infty$. (I know big Picard does say that, but I have trouble visualizing neighborhoods of $\infty$ -- maybe others do too?) 
Obviously if you can find a sequence of longer than 3 such that $f(z_n)$ = w you have a contradiction; so f cannot have an essential singularity at $\infty$.  At that point  it helps if you specifically state that you have a contradiction and what it is; as well as stating that f must look like what? Do you mean a 3rd degree polynomial?  Any old 3rd degree polynomial?  Or as I asked above, could it be a special polynomial of higher degree?
