  This article studies the singular values of entire functions of the form $E^k
(z)+P(z)$ where $E^k$ denotes the $k-$times composition of $e^z$ with itself
and $P$ is any non-constant polynomial. It is proved that the full preimage of
each neighborhood of $\infty$ is an infinitely connected domain without having
any unbounded boundary component. Following the literature, the point at
$\infty$ is called a Baker omitted value for the function in such a situation.
More importantly, there are infinitely many critical values and no finite
asymptotic value and in fact, the set of all critical values is found to be
unbounded for these functions. We also investigate the iteration of three
examples of entire functions with Baker omitted value and prove that these do
not have any Baker wandering domain. There is a conjecture stating that the
number of completely invariant domains of a transcendental entire function is
at most one. How some of these maps are right candidates to work upon in view
of this conjecture is demonstrated.
