An example of a noetherian ring of infinite dimension can be found in Nagata's Local Rings, Appendix A1, Example 1.
Edit: An interesting generalisation of Nagata's construction yielding noetherian rings of infinite dimension whose maximal ideals have prescribed heights was given by Fujita in his article Infinite dimensional Noetherian Hilbert domains, Hiroshima Math. J. 5 (1975), 181–185.
