Does a commutative ring satisfying the following two properties exist?
All ideals are finitely generated;
There are prime ideals with arbitrarily large (finite) minimal generating sets.
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 $n\ge 1$.