High Guardian (archetype)
The High Guardian fighter archetype grants Combat Reflexes as a bonus feat at 2nd level, and allows the use of Strength instead of Dexterity for calculating the number of additional attacks of opportunity.
Cost: Two levels of fighter (if you weren't taking it already), the bonus feats at 1st and 2nd level, and bravery. It also takes your bonus feat at 4th level if you continue with fighter.
Oathbound Paladin (archetype)
The Oath Against Savagery archetype grants an additional number of AoOs per round equal to the character's Charisma bonus while using its Holy Reach ability (which costs 1 use of Smite Evil and lasts for 1 minute).
Cost: Eleven levels of paladin (ouch), divine grace (more ouch), and aura of justice.
Divine Fighting Technique (feat)
The Divine Fighting Technique feat offers a variety of different benefits based on which deity you worship. If you worship Torag, you gain the Way of Patient Strikes benefit, which allows you to add your Wisdom bonus to the number of opportunity attacks you can make. It also allows you to make AoOs while flat-footed and counts as Combat Reflexes for meeting prerequisites.
Cost: A feat, being lawful good, and wielding a warhammer.
Elven Battle Training (feat)
The Elven Battle Training feat grants one additional attack of opportunity per round, as well as +2 to your CMD against disarm & sunder maneuvers.
Cost: A feat, being able to plausibly count as an elf during a census, and wielding a longsword, rapier, or other melee weapon with "elven" in its name.
Line in the Sand (spell)
The Line in the Sand spell lasts 1 round per level and allows the caster to make an additional number of AoOs per round equal to their spellcasting ability modifier against foes within its area (normally a 5' burst around the caster).
Cost: Levels in sorcerer, magus, bloodrager, or wizard (or in a class that lets you poach from their spell lists). One level is enough to get you the spell, but you'll probably want additional levels to increase the duration.
Litany of Warding (spell)
The Litany of Warding spell allows the caster to make two additional attacks of opportunity per round, as well as granting a +2 sacred bonus to AC. It can be cast as a swift action, but unfortunately only lasts for 1 round.
Cost: Ability to cast 2nd-level paladin/antipaladin spells or 3rd-level inquisitor spells (i.e. be one of those classes or steal from their spell list). The spell does not scale in any way with caster level, so once you can cast it there's not much need for more levels in the spellcasting class you got it from.