Since Noetherian rings satisfy the ascending chain condition, every such ring must contain infinitely many chains of prime ideals s.t. the heights of these chains are unbounded.
The only example I know of is the one due to Nagata [1962]: we take a polynomial ring in infinitely many variables over a field, and consider the infinite collection of prime ideals formed by disjoint subsets of the variables. Then we localise the ring by the complement of the union of these prime ideals. With a little work, we can show that by appropriate choice of the subsets, the localised ring will be Noetherian and of infinite Krull dimension. Eisenbud (ex. 9.6) provides a good walkthrough.
The question is: what are other examples of Noetherian rings of infinite Krull dimension?
Best Answer
If you check the book "Krull Dimension" Memoirs of the American Mathematical Society 133 1973, you will find examples of commutative Noetherian integral domains of arbitrary infinite ordinal Krull dimension