Yesterday the moderator Alex Gruber decided to merge this question Example of integral domain with infinitely ascending chain of ideals. which explicitly asks for an example of a non-noetherian ring other than $K[X_1,\dots,X_n,\dots]$ to this one Example of a ring with an infinite inclusion chain of ideals which asks for an example of non-noetherian ring.
As a matter of fact, the question Example of a ring with an infinite inclusion chain of ideals was closed as off-topic for obvious reasons and then deleted. (The example $K[X_1,\dots,X_n,\dots]$ can be found in at least two other older threads on M.SE and at some moment I've provided links to these threads. A new one is Neither Artinian nor Noetherian rings. Also Ideals of a polynomial ring in infinitely many variables which are not finitely generated.) I don't why the moderator has reopened it, but I'd be glad to know his reasons.
While I can agree that this answer could count as a different example from the usual one, the answerer has not done a great job: I can't see any "bi-infinite" (whatever it means) chain of ideals in that ring, and I also can't see a proof showing that the chain (what chain?) is strictly ascending.
On the other side, in the other thread I've posted this answer which satisfied all the requirements: the ring is not noetherian (and this is easily seen) and it's not even isomorphic to $K[X_1,\dots,X_n,\dots]$ (with proof).
Now my answer looks strange: why am I proving that the given example is not isomorphic to $K[X_1,\dots,X_n,\dots]$? The present question asks only for an ascending chain of ideals! (This is why I've to delete it.)
I've posted this in order to find out the moderator arguments for merging the two questions and because my flags telling the above story had no concrete echo so far. (I'm happy that at least some of them were not declined.)