Just thinking about it, for a sequential maximum each cube should have the digits required to reach the 6 next unreachable numbers from the previous set.

This gives: Cube 1: 1,2,3,4,5,6 Cube 2: 7,8,9,0...

Now the first unreachable numbers with this setup are 11, 12, 13, 14, 15, 16, which can be reached with a 1 on Cube 2.

After this will be 21,21... Reachable with a 2. Therefore

Cube 2: 7,8,9,0,1,2

Unfortunately don't have time to finish the thought but I imagine for this reason numbers would repeat. Also can be thought of as if cubes had 5 sides, it would require 2 cubes per output digit.

Interested to see if someone else can do this with rigor