In rings of analytic functions (real or complex), you could take any function with an infinite product expansion, and remove one factor at a time. E.g., you could take $f_0(z)=\sin(\pi z)$, and $f_k(z)=\frac{\sin(\pi z)}{(z-1)(z-2)\cdots(z-k)}$. As indicated in this example, you don't really need to think about infinite products. You can take any function with infinitely many zeros, and divide out by factors corresponding to those zeros (including the multiplicity of each zero, so that taking powers won't add them back in).
(It might be nicer to take $f_0(z)=\sin(\pi z)$ and $f_k(z)=\frac{\sin(\pi z)}{\left(1-\frac{z^2}{1^2}\right)\left(1-\frac{z^2}{2^2}\right)\cdots\left(1-\frac{z^2}{k^2}\right)}$, so that you're actually removing factors from the product expansion.)
You can do this on any connected open subset of $\mathbb{C}$ or $\mathbb{R}$ (connected because you want an integral domain), but another "natural" class of examples that comes to mind is the set of Blaschke products with infinitely many zeros on the open unit disk. In this case, you could take
$$f_k(z)=\frac{f_0(z)}{(\phi_{a_1}(z))^{m_1}(\phi_{a_2}(z))^{m_2}\cdots(\phi_{a_k}(z))^{m_k}},$$
where $a_1,a_2,\ldots$ are the distinct zeros of the infinite Blaschke product $f_0$, $m_k$ is the multiplicity of $a_k$, and $\phi_{a_k}$ is a holomorphic automorphism of the disk that sends $a_k$ to $0$ (unique up to a constant multiple of modulus $1$). Thus each $f_k$ is a Blaschke product and has one less zero than $f_{k-1}$.