Which $AC^0$ problems are not "truly feasible"?
If you want an example of an AC0 function that requires depth $d$, and cannot be computed by AC0 circuits of depth $d-1$, then try the Sipser functions $S^{d,n}$. The superscript $d$ is depth needed for a polynomial-size AC0 circuit. With depth $d-1$, the circuit would need exponentially many gates.

So computing $S^{d,n}$ for $d = 10^{10^{100}}$ would not be "truly feasible."

EDIT: Your question also asks why this would not be feasible. I guess the reason is that $10^{10^{100}}$ is more than the number of atoms in the visible universe.