I have completed the two famous theorems of Picard, presenting their proofs in a graduate course of Complex Analysis, but I have not managed to discover a good number interesting applications.
List of applications (rather straight-forward though):
If a meromorphic function on $\mathbb C$ misses three values, then it is constant.
The equation $f^3+g^3=1$ has non-trivial meromorphic in $\mathbb C$ solutions only if $n\le 3$.
If $f$ is entire and one-to-one, then it is linear.
If $f,g$ are entire and $g'=f(g)$, then $f$ is linear or $g$ is constant.
Could you provide any interesting applications of these theorem?