Does the proof of Picard's theorem become simpler by increasing the number of points that are not attained? Let $f$ be an entire analytic function which attains all but $k$ complex numbers $z_1,\ldots,z_k$. Is there any elementary proof, for some $k$, that $f$ is constant?
 A: There are very many different proofs of Picard's theorem, and some of them are really "simple". (Picard's original proof occupies about 2 lines, using the things already known at that time). None of those proofs that I know simplifies if you assume that the function omits 4, 10, or any finite or countable set of points.
However, if it omits a larger set, for example an
arc, then one can give a "simpler" proof, in the sense that less knowledge is required:
map the complement of this arc onto a bounded region, this can be done by an elementary function, and then apply Liouville's theorem. 
Remark. Let me add that the question whether there exists a "simpler" proof was asked immediately after Picard's proof was published, and it led to a very substantial progress
in several areas of mathematics. People were looking not for a shorter proof but for a "more elementary" proof, using less prerequisite. Many such proofs were found and almost each of them led to a substantial generalization of Picard's theorem. However some very natural generalizations are still unproved. For example, it is conjectured that a holomorphic map from $C$ to $P^2$, the projective space of dimension $2$, omitting a generic curve of degree $5$ must be constant.
A: Suppose you knew that if $f$ is entire and nonconstant, then $f$ attains all but finitely many values.  Then I claim Picard's theorem follows by a completely elementary argument.  To show this, suppose $f$ is a nonconstant entire function that vanishes nowhere.  Then we can write $f=e^g$ for some entire function $g$.  By hypothesis, $g$ must attain all but finitely many values.  It follows that $f$ must attain all values except $0$.
That is, if there were such an elementary proof that you ask for, you would get an equally elementary proof of Picard's theorem.
A: From my personal view, I recommend the paper written by Professor Zhang Guangyuan, the proof  relies some fact that is called  area-length equality.   http://blms.oxfordjournals.org/content/34/2/205.short
