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The Enlightened Magic feature of the Book of Exalted Deeds (DMG, p. 222) allows for 10th-level spell slots per this question.

Nothing seems to disallow level 10 spell slots, but would the spells cast with these slots assume level 10? There are two relevant rules as far as I'm aware. The section on Casting a Spell at a Higher Level says:

When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting. 

However, the section Spell Level says:

Every spell has a level from 0 to 9. 

These seem to contradict if the case of a 10th-level spell slot with Enlightened Magic would allow for level 10 spells considering the definition of spell level.

Since Spell Level is defined between 0 and 9, what happens to a spell's level if you cast a spell with a 10th-level spell slot?

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    \$\begingroup\$ Removed duplicate designation as while the target question may incorporate this one, this question is not entirely identical. Even though a possible answer to the other could include an answer to this one, they are not duplicates. See this meta for more information on my decision. \$\endgroup\$ – David Coffron Feb 1 at 19:29
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Yes, a spell assumes the level of whatever slot is used to cast it

As you quoted:

When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting.

So, if you have something that gives you the ability to cast using 10th level spell slots, then the spells cast using that ability would be considered 10 level spells for that casting.

And, as your previous question addressed, Book of Exalted Deeds does seem to allow the ability to do so when it says:

any spell slot you expend to cast a cleric or paladin spell counts as a spell slot of one level higher.

In this case the fact that the book says:

Every spell has a level from 0 to 9.

is overridden by the fact that the artifact in question allows the spells to be cast using slots higher than that.

As stated in Xanathar's Guide to Everything:

The game also includes elements — class features, spells, magic items, monster abilities, and the like — that sometimes contradict a general rule. When an exception and a general rule disagree, the exception wins. 

So, specific over general, the artifact takes precedent.

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