Just futzing about with terminology a bit.
Die size: The size of an individual die. There is no die size smaller than 1d1, nor higher than 1d20 (percentile dice don't count, they're on their own separate scale). If an effect changes a die size beyond the highest or lowest size, keep track of how many steps beyond the limits it is, but only for interacting with other die size-modifying effects.
1d1 -> 1d2 -> 1d3 -> 1d4 -> 1d6 -> 1d8 -> 1d10 -> 1d12 -> 1d20
Damage Rating: The expression, possibly containing multiple dice, normally used to calculate damage done. Any given damage rating contains only one size of die, although it may involve rolling multiple such dice. There are different tracks of damage rating, each of which has different expressions at each step. If a damage expression it found in multiple damage rating tracks and does not specify which, assume it's in the first such track (as they appear below).
If an effect changes a damage rating beyond the highest or lowest step, keep track of how many steps beyond the limits it is, but only for interacting with other damage rating-modifying effects. However, unlike die sizes, damage ratings extend indefinitely upwards. Every step upward beyond those listed has twice as many dice as the one two steps lower.
If one or more effects modify both a damage expression's die size and damage rating, apply the changes to die size after the changes to damage rating.
Standard (d6): 1d2 -> 1d3 -> 1d4 -> 1d6 -> 1d8 -> 2d6 -> 3d6 -> 4d6 -> 6d6 -> 8d6 -> 12d6 -> 16d6
Advanced (d8): 1d3 -> 1d4 -> 1d6 -> 1d8 -> 1d10 -> 2d8 -> 3d8 -> 4d8 -> 6d8 -> 8d8 -> 12d8 -> 16d8
Compressed (d4): 1d1 -> 1d2 -> 1d3 -> 1d4 -> 1d6 -> 2d4 -> 3d4 -> 4d4 -> 6d4 -> 8d4 -> 12d4 -> 16d4
Expanded (d10): 1d4 -> 1d6 -> 1d8 -> 1d10 -> 1d12 -> 2d10 -> 3d10 -> 4d10 -> 6d10 -> 8d10 -> 12d10 -> 16d10
(Just a note. For common weapon sizes, 1d12 is expanded, 1d10 is advanced, 2d4 is compressed, and everything else is standard.)