I got curious how the upgrade station decides whether an upgrade works or the item breaks, so I went through the game code and files.
The roll is not random. For a new character, the first roll on each item and level comes from the character’s name. A character named X always gets the same results on the same item. A character named Y gets different ones.
So I built a name finder. You enter an item and the quality it starts at, and it lists character names that give exactly 8 upgrades before the next attempt breaks it. 8 upgrades is the most one item can get this way. But multiple items can reach higher levels.
Name Finder: https://melichor.github.io/ValheimSaveParser/ (scroll down to “Upgrade name finder”)
Check the notes that are listed there, some are very important to what you're trying to achieve.
Simple mode just gives you a name with 8 upgrades, advanced mode allows you to search for paths with multiple items which can reach really high upgrade levels.
EDIT:
I HAVE BROKEN IT FURTHER.
I have figured out, that the RNG is actually not tied to the item, but only to the character.
That means that you can chain these upgrades, I have updated the tool