Let $X$ be the space of entire functions on $\mathbb{C}$ endowed with the topology of uniform convergence on compact sets. Let $a$ be a nonzero complex number. Let $T: X\to X$ be defined by $T(f)(z)=f(z+a)$. By a result of Birkhoff (see Dynamics of Linear Operators, by F. Bayart and E. Matheron, p. 3), there exists an $f\in X$ such that $O(f,T)=\{T^nf: n=0,1,2,\cdots\}$ is dense in $X$.
Does anyone know a concrete example of such $f$?
On
Except the Riemann Zeta function and similar functions (which are not entire functions), no concrete example is known so far.
For a survey of known results, I suggest you take a look at the recent papers by Paul Gauthier, e.g. this one. After Theorem 13, there is a remark saying that no concrete example of universal entire functions is known.
Such functions are called universal. The Riemann Zeta function was the first function to be known to have universal properties. See this link and these slides. The right formulation of Voronin theorem is given in the latter one. There is a typo in the former one: $f(z)$ should be instead of $\zeta(z)$.