I'm trying to find any and every way I can to get a Wizard in Pathfinder to learn more spells every time they level up.
I remember quite a few from D&D 3.X but I don't know any for Pathfinder.
Wizard: Add one spell from the wizard spell list to the wizard’s spellbook. This spell must be at least one level below the highest spell level the wizard can cast.
Similarly, Fetchling wizards may do the same, but only with Illusion (shadow) spells.
Wizard: Add one spell from the wizard spell list to wizard’s spellbook. The spell must be at or below the highest level the wizard can cast and be of the illusion (shadow) subschool or have the darkness descriptor.
Undine may also add additional spells, but taken from the wizard, druid or cleric spell list and must have the [Water] descriptor.
Add one spell from the cleric, druid, or wizard spell list with the water descriptor to the wizard’s spellbook. This spell must be at least one level below the highest spell level the wizard can cast. This spell is treated as one level higher unless it also appears on the wizard spell list.
Pact Wizards may add their Patron's spells to their spellbook when they level up.
Poleiheira Wizards add four spells instead of two to their spellbooks every time they gain a new wizard level.
There are also a few archetypes that add specific spells to their spellbooks on specific levels, like the Undead Master.