In this question, I gave an example of a ring whose lattice of two-sided ideals is order-isomorphic to $\omega+1$. I've been playing a bit with trying to find rings with a given lattice of ideals since, and one case I found interesting and difficult at the same time. (Difficult for me of course. I've learned here that many questions I can't answer turn out to be trivial.)
Is there a unital ring whose lattice of two-sided ideals is order-isomorphic to $(\omega+1)^{\operatorname{op}}?$
Maybe you can tell me if this is what you're looking for.
What about the power series $R=\mathbb{F}[[x]]$? Doesn't $R\supset(x)\supset(x^2)\supset\dots$ make the chain you are looking for?