When iterating transcendental entire functions, it is possible to have multiply connected Fatou components. The earliest examples of such functions occur as infinite products. In the first part of this talk, I will show how to construct such an infinite product and discuss what we know about the dynamics and geometry of such domains. In the second part, I will discuss a new approach for constructing such functions, along with some applications. This talk features joint work with Kirill Lazebnik.