The subclass description says "The spells you learn at 8th, 14th, and 20th level can come from any school of magic." But the chart does not show any spells gained at these levels? Or is that just the spell slots don't increase?
It is your spell slots that are not changing at 8th, 14th, and 20th levels. Your number of spells known increases from 5 to 6, 9 to 10, and 12 to 13 at those levels respectively.
If interested here's the spell learning curve for Arcane Trickster Rogue laid down by level:
Level 03: 2 new 1st-level wizard spell from the enchantment or illusion school, 1st level wizard spell from any school, mage hand and two other wizard cantrips
Level 04: 1 new 1st-level wizard spell from the enchantment or illusion school
Level 05: No new spell learned
Level 06: No new spell learned
Level 07: 1 new 1st or 2nd level wizard spell from the enchantment or illusion school
Level 08: 1 new 1st or 2nd level wizard spell from any school
Level 09: No new spell learned
Level 10: 1 new 1st or 2nd level wizard spell from the enchantment or illusion school and 1 wizard cantrips
Level 11: 1 new 1st or 2nd level wizard spell from the enchantment or illusion school
Level 12: No new spell learned
Level 13: 1 new 1st, 2nd or 3rd level wizard spell from the enchantment or illusion school
Level 14: 1 new 1st, 2nd or 3rd level wizard spell from any school
Level 15: No new spell learned
Level 16: 1 new 1st, 2nd or 3rd level wizard spell from the enchantment or illusion school
Level 17: No new spell learned
Level 18: No new spell learned
Level 19: 1 new 1st, 2nd, 3rd or 4th level wizard spell from the enchantment or illusion school
Level 20: 1 new 1st, 2nd, 3rd or 4th level wizard spell from any school
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The subclass description says "The spells you learn at 8th, 14th, and 20th level can come from any school of magic." But the chart does not show any spells gained at these levels? Or is that just the spell slots don't increase?
It is your spell slots that are not changing at 8th, 14th, and 20th levels. Your number of spells known increases from 5 to 6, 9 to 10, and 12 to 13 at those levels respectively.
If interested here's the spell learning curve for Arcane Trickster Rogue laid down by level:
Level 03: 2 new 1st-level wizard spell from the enchantment or illusion school, 1st level wizard spell from any school, mage hand and two other wizard cantrips
Level 04: 1 new 1st-level wizard spell from the enchantment or illusion school
Level 05: No new spell learned
Level 06: No new spell learned
Level 07: 1 new 1st or 2nd level wizard spell from the enchantment or illusion school
Level 08: 1 new 1st or 2nd level wizard spell from any school
Level 09: No new spell learned
Level 10: 1 new 1st or 2nd level wizard spell from the enchantment or illusion school and 1 wizard cantrips
Level 11: 1 new 1st or 2nd level wizard spell from the enchantment or illusion school
Level 12: No new spell learned
Level 13: 1 new 1st, 2nd or 3rd level wizard spell from the enchantment or illusion school
Level 14: 1 new 1st, 2nd or 3rd level wizard spell from any school
Level 15: No new spell learned
Level 16: 1 new 1st, 2nd or 3rd level wizard spell from the enchantment or illusion school
Level 17: No new spell learned
Level 18: No new spell learned
Level 19: 1 new 1st, 2nd, 3rd or 4th level wizard spell from the enchantment or illusion school
Level 20: 1 new 1st, 2nd, 3rd or 4th level wizard spell from any school