If $A$ is a commutative ring we have the estimate $$ \dim (A)+1 \le \dim (A[x])\le 2\dim (A)+1 $$ for the Krull dimension, with $\dim (A)+1 = \dim (A[x])$ for Noetherian rings. I am looking for nice examples of rings $A$ so that $A[x]$ has Krull dimension $\dim (A)+2, \dim(A)+3,\ldots ,2\dim(A)+1$.
Your question is essentially completely answered in the paper The dimension sequence of a commutative ring by Arnold and Gilmer (Amer. J. Math. 96 (1974), 385408). EDIT: The aforementioned paper by Arnold and Gilmer treats the case of an arbitrary finite number of variables. Since you are interested only in the case of one variable, the general but still rather concrete construction of rings with the desired properties given in Bourbaki's Algèbre commutative VIII.2 Exercice 7 should be sufficient. 

