Here is a slightly different version of   Aakumadula's answer to my question.

Let's say that $f$ omits $g$ if $f(z)$ is never equal to $g(z)$.

Let $G$ be the group of fractional linear transformations such that the unit disc $U$ modulo $G$
is C\ { 0,1 } . (Can anyone suggest a short and recognizable name for this group??
This is a truly fundamental object of complex analysis, and the shortest name for it
that I know is the "principal congruence subgroup of level 2 of the modular group".
Sounds scary for many people).  

Proposition. TFAE: There exists $f$ holomorphic in $U$ that omits $0,1,\infty$ and $\lambda(z)$,
and: there exists $f: U\to U$ that omits all elements of $G$.

 Proof. Evident.

Now it is well-known that there exists $f$ holomorphic in $U$ which omits $0,1,\infty$ and $\lambda$.
This is by "extension of holomorphic families of injections" of Slodkowski.

The theorem of Slodkowski says that whenever you have any number of holomorphic functions in $U$
with disjoint graphs, you can add one whose graph is disjoint from those given functions.
And even prescribe the value of this one added function at one point.