Well for starters, we can get to 9 domains with an 11th level character.
Cleric 1 / Druid 1 / Hunter 3 / Paladin 4 / Occultist 1 / Medium 1.
At first level, a chooses two domains granted by their deity. There are some alignment-based restrictions, but otherwise the choices are limited by the deity you chose.
A 1st level druid gets the Nature Bond class feature, and choose either an animal companion or a cleric's domain. The choices are limited, although some of the splat books (Ultimate this, Ultimate that) enable more domain choices.
Just kidding! If you have levels in cleric, then you can't dip into Inquisitor to get an extra domain.
If the inquisitor has cleric levels, one of her two domain selections must be the same domain selected as an inquisitor. Levels of cleric and inquisitor stack for the purpose of determining domain powers and abilities, but not for bonus spells.
At 3rd level, this Hunter archetype lets them choose an additional domain. Unlike the inquisitor, there are no restrictions if you multiclass.
At 3rd level, a divine hunter learns to call upon the power of her deity. The divine hunter must select one domain from those available to her deity. She gains the granted powers of this domain, using her hunter level – 2 as her cleric level for determining when the powers are gained and what effects they have. Once she chooses this domain, it cannot be changed.
Either option involves a 4th level paladin. Sacred Servant lets you pick a cleric domain and get the additional spells, whereas the Temple Champion lets you pick a cleric domain and get the 1st level domain power. Both seem to qualify here.
Occultists are weird, and this one is weirder because it's also a divine caster.
A reliquarian gains access to one of her deity’s domains. She treats her occultist level as her cleric level for the purposes of access to domain powers and their strength. Additionally, the reliquarian adds the domain’s spells to her occultist spell list and her spells known.
Even weirder than the Occultist, this Medium archetype can choose a different domain each day. There are plenty of options. You gain the domain powers and the domain spells.
While channeling a wendo, the wendo caller gains the 1st-level domain power of one druid animal or terrain domain, or the 1st-level domain power of one of the following cleric domains: Air, Animal, Community, Darkness, Death, Destruction, Earth, Fire, Healing, Knowledge, Protection, Rune, Strength, Sun, Trickery, War, Water, or Weather.
The wendo caller has an effective cleric or druid level equal to 1/2 his medium level (minimum 1st) for the purpose of this ability. In addition, he adds all spells from his chosen domain that are of a spell level he can cast (including modifications to the wendo caller’s spellcasting from the archmage arcana or divine surge lesser spirit powers) to his medium spell list and medium spells known as medium spells of their domain spell level.
The Advanced Class Guide introduced Believer's Boon, a loosely worded feat. Pick a deity, get one of their domains. There are no restrictions if you're already a cleric. Plus, you can take this feat at 1st level and again at 3rd level, for a sum total of two additional domains.
When you take this feat, choose one domain granted by your deity. You can use the 1st-level domain ability that clerics of that domain can use a number of times or rounds per day, but you can use it only once per day or 1 round per day, whichever is appropriate. Your effective cleric level in regard to this ability is 1st level. If the domain has a 1st-level ability that does not meet this specification, you cannot use it.
Special: You can take this feat twice. When you take it a second time, you choose another domain granted by your deity.
Total = 2 + 1 + 1 + 1 +1+ 1 + 2 = 9
Not included, because they aren't "Domains":
- Warpriest's Blessings: Warpriest chooses two of their deity's domains, but they get Blessings instead of domain abilities.
- Ranger (Divine Tracker)'s Blessings: Same problem as warpriest. You pick a deity's domains, but get warpriest minor blessings.
- Wizard's Arcane School: Like a domain, it grants limited abilities, and an additional spell per day per level.
- Shaman's Spirits: Like a domain, it grants limited abilities, and an additional spell per day per level. Also a divine caster.
- Witch's Patrons: Like a cleric, witches get their powers from a patron, which also gives them additional spells known, but not additional spells per day.
- Divine Scion's Domain Specialization: Pick a domain, and gain a spell-like ability. It doesn't function like a cleric domain.
- Bard (Faith Singer)'s Devout Spell Knowledge: Pick a domain, and gain some spell-like abilities. Doesn't function like a cleric domain.