Stepping in here folk; please focus on the users original question and not get too focused on typing styles.
As for the original question remember the general rule 'Specific beats general'.
For example for Cleric, it explains that the table shows the maximum number of prepared spells for that level. For example a 4th Level Cleric can prepare up to 7 spells.
It also then has: (Emphasis mine)
If another Cleric feature gives you spells that you always have prepared, those spells don’t count against the number of spells you can prepare with this feature, but those spells otherwise count as Cleric spells for you.
The Subclass Life Cleric gives domain spells (Emphasis mine):
When you reach a Cleric level specified in the Life Domain Spells table, you thereafter always have the listed spells prepared.
So our 4th Level Cleric can prepare up to 7 spells, but also always has Aid, Bless, Cure Wounds, Lesser Restoration prepared, making their total prepared spells for that day 11.
So it's not that the 'Maximum prepared spells' changes with subclass, but that there are certain prepared spells that do not count towards this total and are always prepared.
There is no such thing as a "Maximum Prepared Spells" and as such there is nothing that increases it. The spellcasting feature and some other features/feat grant a number of prepared spells but these generally do not increase the number granted by another feat or feature.
The number you can choose for yourself? No. Your total number prepared can; various subclass features, feats, and a few class features give you additional prepared spells that don't count against your total from class level(s).
This is technically incorrect as some features/feats do grant spells "you can choose for yourself"
Magic Initiate Feat - you can choose 2 cantrips and 1 spell, which you can change one of when you level-up
Pact of the Tome - you get 2 1st level ritual spells which you can change each time you summon the Tome
College of Lore Bard - Magical Discovery - grants you two additional spells which can be from the Bard, Cleric, Druid or Wizard spell lists, these can be changed when you level-up.
None of these count against your number of prepared spells but they are spells you can choose yourself from a list.
Stepping in here folk; please focus on the users original question and not get too focused on typing styles.
As for the original question remember the general rule 'Specific beats general'.
For example for Cleric, it explains that the table shows the maximum number of prepared spells for that level. For example a 4th Level Cleric can prepare up to 7 spells.
It also then has: (Emphasis mine)
The Subclass Life Cleric gives domain spells (Emphasis mine):
So our 4th Level Cleric can prepare up to 7 spells, but also always has Aid, Bless, Cure Wounds, Lesser Restoration prepared, making their total prepared spells for that day 11.
So it's not that the 'Maximum prepared spells' changes with subclass, but that there are certain prepared spells that do not count towards this total and are always prepared.
D&D Beyond ToS || D&D Beyond Support
There is no such thing as a "Maximum Prepared Spells" and as such there is nothing that increases it. The spellcasting feature and some other features/feat grant a number of prepared spells but these generally do not increase the number granted by another feat or feature.
This is technically incorrect as some features/feats do grant spells "you can choose for yourself"
Magic Initiate Feat - you can choose 2 cantrips and 1 spell, which you can change one of when you level-up
Pact of the Tome - you get 2 1st level ritual spells which you can change each time you summon the Tome
College of Lore Bard - Magical Discovery - grants you two additional spells which can be from the Bard, Cleric, Druid or Wizard spell lists, these can be changed when you level-up.
None of these count against your number of prepared spells but they are spells you can choose yourself from a list.