  Let $H(\mathbb{C})$ be the set of all entire functions endowed with the
topology of uniform convergence on compact sets. Let $\lambda,b\in\mathbb{C}$,
let $C_{\lambda,b}:H(\mathbb{C})\to H(\mathbb{C})$ be the composition operator
$C_{\lambda,b} f(z)=f(\lambda z+b)$, and let $D$ be the derivative operator. We
extend results on the hypercyclicity of the non-convolution operators
$T_{\lambda,b}=C_{\lambda,b} \circ D$ by showing that whenever $|\lambda|\geq
1$, the collection of operators \begin{align*}
  \{\psi(T_{\lambda,b}): \psi(z)\in H(\mathbb{C}), \psi(0)=0 \text{ and }
\psi(T_{\lambda,b}) \text{ is continuous}\} \end{align*} forms an algebra under
the usual addition and multiplication of operators which consists entirely of
hypercyclic operators (i.e., each operator has a dense orbit). We also show
that the collection of operators \begin{align*}
  \{C_{\lambda,b}\circ\varphi(D): \varphi(z) \text{ is an entire function of
exponential type with } \varphi(0)=0\} \end{align*} consists entirely of
hypercyclic operators.
