Given a finite dimensional non-Gorenstein algebra $A$, do we have $Ext^i(D(A),A) \neq 0$ for infinitely many $i$? (We can assume A is local or commutative if that helps).
All I can show is that such algebras would be pretty exotic when this holds only for finitely many i.



Question on Ext for finite dimensional algebras
My modules are right modules.


Let $M$ be a non-projective module for a self-injective algebra $B$ such that $\text{Ext}^i\_B(M,M)=0$ for $i>t$ (i.e., what you've called a "strange" module in some recent questions). Let
$$A=\pmatrix{k&M\\0&B}.$$
Then $A$ is not Gorenstein, as it has a one-dimensional injective module with infinite projective dimension, and according to my calculations, $\text{Ext}^i\_A(DA,A)=0$ for $i>t+1$.


An $A$-module is a "row vector" $\pmatrix{V&X}$, where $V$ is a vector space and $X$ a $B$-module, together with a $B$-module map $V\otimes\_kM\to X$.


$DA$ is the direct sum of two injectives $I\_1=\pmatrix{k&0}$ and $I\_2=\pmatrix{DM&DB}$ corresponding to the columns of $A$. 


If 
$$\dots P\_1\to P\_0\to M\to0$$ 
is a $B$-projective resolution of $M$, then
$$\dots\pmatrix{0&P\_1}\to\pmatrix{0&P\_0}\to\pmatrix{k&M}\to\pmatrix{k&0}\to0$$
is an $A$-projective resolution of $I\_1$, and so 
$$\text{Ext}^{i+1}\_A(I\_1,A)=\text{Ext}^i\_B(M,M\oplus B)$$
for $i>1$, which is zero for $i>t$ by the assumptions on $B$ and $M$.


There is a short exact sequence
$$0\to\pmatrix{0&DB}\to I\_2\to\pmatrix{DM&0}\to0.$$
The last term is a direct sum of copies of $I\_1$, and the first term is a projective $A$-module, so the $\text{Ext}^\*\_A(-,A)$ long exact sequence gives $\text{Ext}^i\_A(I\_2,A)=0$ for $i>t+1$.


