Does an integer appear in the last two digits (in decimal representation) of infinitely many $a_{i}$?
Since $\varphi(16)=8$, we have $a_{i-2}=3^{a_{i-3}} \equiv 3^{3} \equiv 11(\bmod 16)$.
Since $\varphi(40)=16$, we have $a_{i-1}=3^{a_{i-2}} \equiv 3^{11} \equiv 27(\bmod 40)$.
Finally, since $\varphi(100)=40$, we have $a_{i}=3^{a_{i-1}} \equiv 3^{27} \equiv 87(\bmod 100)$.
Thus, only 87 appears in the last two digits of infinitely many $a_{i}$.