To my knowledge, there's no rule stating a minimum AC. I've never heard of an AC in the negatives, but it'd be pretty amusing to see such a build, if it is possible.
Build with negative AC (while raging it's at -1).
There is no negative cap.
An AC below 10 is possible, see this example in Paizo's NPC Codex:
Paizo usually describes minimum (and maximum) values carefully. So if it's not written, it doesn't exist. Armor class section in Core rulebook page 179 doesn't describe any minimum, so the barbarian has to deal with his low AC...
Counter example:
There are a lot of modifiers that can push your AC below 10. Low Dex, size, being paralyzed, etc.
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Many oozes and the like have ACs around 5.
No, there is no minimum AC
Got ya trumped: Gelatinous Cube, our old iconic monster, has an AC of 4.
As a corollary, it is also possible to have fairly large Initiative penalties, like a Cleric wearing full plate and using a tower shield, both without proficiency:
Full plate: -6 ACP
Tower Shield: -10 ACP
Dex 7: -2 Dex mod
Gives an Init mod of -18, so last, dead last, or probably last, with no better than a 2 (20-18) as your initiative count...
Of course, that leaves your flatfooted AC equal to your normal AC, both at 21 before magical enhancements, or other AC items, and touch as a lovely 8...
As an add-on, there is a cracked or flawed Ioun stone that gives a +2 Perception bonus, at the cost of a (minor) -1 penalty to Initiative.
Got ya trumped: Gelatinous Cube, our old iconic monster, has an AC of 4.
Carnivorous blob, AC 2. Great wyrm red dragons have a touch AC of 0, while an aspidochelone has a touch AC of -1.
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Of course, the bloodrager with a total AC of -5 NEEDS a lot of hit points and even more healing to survive, because he's getting hit on a 2+ by nearly everything. So while there's no minimum AC, you probably shouldn't work to get one as low as you possibly can.
On the other hand, there's no maximum AC either. The only real soft cap on AC is that there's only so many items you can acquire to boost it. With idealized gear I threw together some AC comparisons for one character I play as comparisons to see what worked better for self sufficient AC. You know, cause I was experimenting.
Lev 20 hydrokinetic
+5 mithral chain shirt 9ac
+5 ring of protection 5ac
Amulet of Natural Armor +5 5ac
Improved Natural Armor x3 4ac
Shroud of Water (shield) +3 burn 9ac
Dodge 1ac
Dex 16 3ac
Base 10 ac
Total Armor Class 46
Shroud of Water (armor) +3 burn 12 AC
ring of shielding 2ac
Total Armor Class 42
Level 10 Hydrokinetic/Level 10 Monk
As Above -chain shirt
Shroud of Water (armor) +3 burn 9ac
Ring of Force Shield 2ac
Monk 2ac
Wisdom 18 4ac
Dex 16 3ac
Total Armor Class 45 (49 with Ki)
Water Shround (shield) +2 burn 6ac
Bracers of Armor +8
Total AC 48 (52 with ki)
So as you can see, while there's no cap on AC your total can vary depending on what gear you acquire and what abilities you're relying on.
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Or you could have a level 12 Druid/level 8 Half Orc Monk Ancient Guardian w/ Defense Subdomain
Starting Dex of 18, Wis of 16
Diminutive Bat Form
Base 10
Bracers of Armor +8 (Put on after Wild Shape)
Ring of Protection +5
Barkskin +5
Jingasa +1
Dusty Rose Ioun Stone +1
Fates Favored +1
Dodge +1
Dex 40 (18 base, +5 level, +5 Book, +6 Enhancement, +6 Bat form) +15
Wis 26 (16 base, +4 Book, +6 Enhancement) +8
Size +4
Natural Armor +5 (+1 Base, +4 Half Orc favored Class Bonus)
Monk Lvl +3 (Monk's Robes)
For a 67 +
Fight Defensive +4
Combat Expertise +5
Cast Shield +4
For a total of AC 80 (84 w/Ki)