We have: 

111³ = 1367631 have 7 digits, remove

222³ = 10941048 , remove

333³ = 36926037, satisfy

A > 3 => \(\left(AAA\right)^3=\overline{X...........}\) with:

* X > 7 when (AAA)³ have 7 digits

* X < 9 when (AAA)3 have 8 digits 

So A = 3