Noetherian ring whose ideals have arbitrarily large number of generators

Solution 1:

The example of Nagata of a noetherian ring with infinite Krull dimension satisfies your requirements: all ideals are finitely generated and prime ideals must have arbitrarily large number of generators, otherwise their heights would be bounded, contradiction.

Edit. After posting this answer I wondered if a noetherian ring containing ideals with arbitrary large minimal generating sets has necessarily infinite Krull dimension. The answer is NO and this paper provides examples of prime ideals of height $2$ in the power series ring $K[[X,Y,Z]]$ having at least $n$ generators for all $n\ge 1$.