The DMG states
"A creature can be attuned to no more than three magic items at any given time"
Let's interpret this as
"A creature can be attuned to no more than three powerful magic items at any given time"
and add
"A creature can attune to two items of lesser power in place of attuning to one powerful item".
Then, all we need to do is define "powerful" and "lesser power". I suggest tying this to the
tiers of play. And the DMG already contains all the guidance we need - on page 135.
All this means:
- For a tier I character (level 1-4) a powerful magic item is Rare or higher. Common and uncommon items are considered to be of lesser power to you.
- For a tier II character (level 5-10) a powerful magic item is Very Rare or higher. Common, uncommon and rare items are considered to be of lesser power to you.
- For a tier III character (level 11-16) a powerful magic item is Legendary or higher. Common, uncommon, rare and very rare items are considered to be of lesser power to you.
- For a tier IV character (level 17+) only Artifacts count as powerful magic items. Common through legendary items are considered to be of lesser power to you.
So a tenth level character could have three Very Rare items attuned, for example. Or one Very Rare, one Legendary, and one Artefact - items of those rarities all count as "powerful" for a tier II character.
This houserule does not prevent you the DM from giving a 1st-level character that Ring of Invisibility!
Alternatively, she could give up one of those items to attune two Rare items instead. Or she could give up on "powerful" items altogether, and instead attune up to six Rare and Uncommon items. At least until she levels into the next tier of play!
As you can see, the number of items is the same as the proficiency bonus suggestion: six items at the most. But. There's a trade-off to be made. A (hopefully interesting) choice. For all levels, not just the highest ones.