Wizards of the Universalist school gain an ability at 8th level, Metamagic Mastery, which allows them to apply the effects of one metamagic feat without increasing the casting time or altering the spell level. This ability can be used once per day at 8th level, then an additional time per day every 2 additional wizard levels (ex: 16th level, 5 times per day).
Also, the Pact Wizard archetype is able to reduce the increase from metamagic feats by 1 spell slot, when using their Great Power, Greater Expense at 15th. This has a limited use per day (based on her int), but also has other benefits, such as being able to roll twice and pick the best result for caster level, concentration, initiative checks, or saving throws.
11th level Psychics have access to a Major Amplification called Mimic Metamagic, which allows them to mimic the effects of two chosen metamagic feats when casting their psychic spells by using points from their phrenic pool. This does not increase the casting time of the spell, nor the spell level necessary to cast the spell.
I know you didn't ask for magic items, but Im listing those here for completeness. They are the next easiest way to accomplish that without using optional rules.
Metamagic Gems are single-use magic items that allow you to apply a metamagic feat's effect to a spell when it's being cast without altering the spell level. This means that a Maximizing Sapphire can be used to cast a maximized Fireball that will still use up a 3rd level slot (assuming sorcerer/wizard).
Similarly, Metamagic Rods will allow your character to apply the metamagic effect on the rod on any spell, depending on the level of the rod, without altering the spell's level, but only for a limited number of times per day. A lesser Maximize Metamagic Rod would allow you to maximize up to 3 different spells from levels 1st to 3rd without increasing the spell level, which would recharge the next day.
However, there are even more specific items, that apply the effects of a single metamagic feat to certain spells you cast, like the Book of Extended Summoning.
Mythic Adventures introduced Legendary items, which have an option to be enchanted with Metamagician, which allows you to spend some of the item's Legendary power to fuel a metamagic effect you know without increasing the cast time or spell level.
Both the Hierophant and the Archmage have abilities that allows them to cast spells using some of their mythic power instead of spending the spell slot.
For Hierophants, they use the Inspired Spell ability and may cast any spell on their list, even if not memorized, without spending a spell slot and are allowed to apply any metamagic feat they know, as long as it doesn't alter the spell level beyond what you are allowed to cast.
For Archmages, they have a similar ability, Wild Arcana, but they also have another ability that can be picked at 3rd tier, Arcane Metamastery, which allows them to pick one metamagic feat they know, as long as it only increases the spell level by at most one level (which increases every time they pick this ability), and apply that metamagic feat to any arcane spell they cast for the next 10 rounds.
One of the examples of Wild Magic effects that a spell cast can do using the (optional) Wild Magic rules from Pathfinder Unchained is to allow a spellcaster to make a caster level check with a DC of 10 + spell level + 5 per level increase of the metamagic feat they want to use, which then allows casting the spell without increasing the casting time or using a higher level slot. But if she fails on the check, they roll on the wild magic surge tables.