The model

The model contains 3 parts:

The first tier; necessary to print as the base and to store 1 set of 7 dice

The lid; necessary to close the box

The second tier; add-on part to store an additional set of 7 dice.
The second tier can snap between the first tier and the lid, but two of the second tier parts can also snap together, which means that you can stack them as high as you want!

How to print

The first tier and the lid can be printed without supports,

The second tier should be printable without supports if your part cooling is good enough, and if you set the “top/bottom pattern” to concentric. To be safe you can still enable supports for this part though. 

Also be sure to orient the lid part with the logo facing upwards.

I actually recommend printing these parts at a rather high layer height, as the model doesn't have a lot of fine details, and it greatly reduces the print time.

I designed the box based on a set of dice I have where the biggest die (the d20) is 20mm tall, be sure to adjust the size of the box in your slicer if you are printing it for larger dice.