I may be wrong but im pretty sure Animal Companions dont work like that. Whatever animal companion you have is the animal companion for all the classes unless the class forces you to change the companion(Like The Mammoth Rider Prestige Class). To get multiple animal companions you only need to go the Pack Lord archtype for druids. You can get a new companion at every level.
I originally was going to take both the Sacred Huntmaster Archetype and the Hunter in a multiclass build. Then I saw the Mad Dog Archetype for Barbarian.
If for some not-optimal reason I did this, what would my animal companions be rated at for a Level 2 Mad Dog Barb., level 3 Sacred Huntsmaster and level 3 Hunter? Character level 8 with all those 3 classes.
Animal Companion (Ex): At 1st level, a hunter forms a
bond with an animal companion. A hunter may begin play
with any of the animals available to a druid. This animal
is a loyal companion that accompanies the hunter on her
adventures. This ability functions like the druid animal
companion ability (which is part of the nature bond class
feature). The hunter’s effective druid level is equal to her
hunter level. If a character receives an animal companion
from more than one source, her effective druid levels stack
for the purposes of determining the companion’s statistics
and abilities.
The answer to the thread's title is: one. All animal companions (including the paladin and cavalier mounts) include verbage that basically says the following: "effective druid levels from multiple classes stack to determine your animal companion's abilities."
So yes. At Level 8, a Barbarian (Mad Dog) 2 / Inquisitor (Sacred Huntsmaster) 3 / Hunter 3 would be considered an 8th Level Druid when determining her animal companion's abilities.
That said, this combo really isn't worth it. Note that your Inquisitor archetype doesn't get Solo Tactics; she trades it for Hunter's Tactics, the same ability given to you by the 3rd Level Hunter ability. They're redundant with one another. Not to mention you'd have subpar spellcasting in two different classes. Plus your spellcasting is going to be that of two 2nd-level characters and your base attack bonus is lagging a bit behind too.
To answer your question about the Mad Dog, the archetype doesn't care what type of animal companion you have. Take a dinosaur if you want it.
The inquisitor doesn't lose Solo Tactics to gain Hunter's tactics.
There are a few archetypes like Packmaster that let you get multiple animal companions, but you end up splitting the levels between them. It's worth noting that in PFS you're limited to one "combat" animal, so even if you ended up getting an animal companion, a familiar, and an eidolon, only one could fight with you (for this reason, Packmaster isn't legal in PFS).
To answer the original question: be a Pack Lord druid, and you can have as many animal companions as you have levels.
Sure, they might all be 1st-level companions, but I think that's how you would get the max number. I've thought about making a Pack Lord that has a flock of birds, or maybe a pack of bloodhounds so you could roll a handful of Survival checks to track by scent. The issue, of course, is protecting them...
The inquisitor doesn't lose Solo Tactics to gain Hunter's tactics.This is correct.
Guess I need to find the section where it says you can only get 1 animal companion. Figured I could get more when browsing these 3 class types.
Well then Barbarian Mad Dog is out then. Still want the Sacred HM and Hunter classes together.
The inquisitor get's both solo tactics and hunter's tactics.
Also, what is it you're wanting to do? I'm getting confused as to what you're asking about and trying to do now.