Elaborating on what Gerhard Paseman points out, selecting sets $A_i = \{kh^i \mid 0 \le k < h\}$ will work.

More generally, if you define your sets inductively (and restrict to only using positive integers) so that the smallest pairwise difference in elements in $A_i$ is greater than the largest element in $A_1 + \cdots + A_{i-1}$ then it will also avoid any collisions, and have $|A_1 + \cdots + A_{i}| = |A_1| \cdots |A_i|$.

Edit to provide example:

For example, choosing $A_1 = \{0,1,4\}$ and then $A_2$ any set with elements at least $5$ apart $A_2 = \{0,7, 22, 54, 59\}$ and now $A_3$ any set with elements at least $64$ apart $A_3 = \{0, 77, 200\}$ and $A_4$ having elements at least $264$ apart, etc. You can continue at will.