
I'm pretty sure this can't be done in real base 10. The closest I can get is:
0, 1, 2, 3, 4, 5
0, 1, 2, 6, 7, 8, 9
The problem is that the first digit can be 0, 1, or 2 (or 3, but that's a special case). So what we end up with is we must have 3 on the opposite die as 0 and 1. But we also need to be able to reproduce 0*, 1* and 2*, which means each of those digits must be on both dice. Sadly, using base 10 with two 6sided dice only allows us to duplicate 2 numbers.
Therefore, I'm assuming the first digit may be absent to signify 0*:
0, 1, 2, 4, 5, 6
1, 2, 3, 7, 8, 9
This allows combinations such as "1", "30", and so forth. The first number that can't be created is 33.
There are also several ways to do it outside base 10 (using base 6, 7 or 8) with larger contiguous segments, but I'm guessing by your wording that this isn't what you were looking for. However, I'll give some anyway. For all of these solutions I'm assuming I can't do the missing digit trick, which is why I'm not allowing 9 (it requires the missing digit); if the missing bit were allowed, then we'd just stick with base 10 anyway.
Base 6:
0, 1, 2, 3, 4, 5
0, 1, 2, 3, 4, 5
Base 7:
0, 1, 2, 3, 4, 5
0, 1, 2, 3, 4, 6
Base 8:
0, 1, 2, 3, 4, 5
0, 1, 2, 3, 6, 7
Base six allows us to go up to 35 (55 in base six), base seven up to 40 (55) and base eight up to 36 (44). In the end, this makes base seven our "optimal" solution, but it's not intuitive for most people to work with. On the other hand, base eight is quite common in the realm of computer science and it happens to be the next best solution, so why not?
So, do I win anything?

