Growing digits

Logic Level 1

$\begin{array} { rrrrr } & & & & & A \\ & & & & A & A \\ & & & A & A & A \\ + & & A & A & A & A \\ \hline & \large\square & \large\square & \large\square & \large\square & \large\square \\ \end{array}$

The above is an incomplete cryptogram, where $A$ is a positive, single-digit integer. As shown, the final sum is a 5-digit number.

What is this final sum?