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The exact example I am using is a Headband of Vast Intelligence and the spell Fox's Cunning and the headband and Crimson Sphere Ioun Stone

Both combinations grant an enhancement bonus thus normally wouldn't stack. However after 24 hours the Headband's bonus becomes a permanent bonus as long as it is not removed.

Does this change the bonus the headband grants in anyway to allow the other spell/item to also grant the bonus or does it still count as an enhancement bonus meaning the others cannot stack?

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The enhancement bonus part does not stack with the other spells or items. Whichever bonus is the largest at the time is the one used.

No matter if the enhancement bonus from the headband is being used or not, the skill bonus from the headband persist so long as the headband is worn.

So even if you have a +4 from a Fox's cunning spell, you would still get the skill bonus from the +2 Headband even if you are not getting the +2 INT from the headband.

Also, despite the bonus to INT being permanent after 24 hours, skill ranks are not increased retroactively. http://paizo.com/paizo/faq/v5748nruor1fm#v5748eaic9og7

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    \$\begingroup\$ Note that the INT permanent thingie, it's just to prevent abusing the ability to share items among the party. If it was considered a permanent increase immediately, you could have a headband of intellect used by the wizard to get a bonus spell per day, have him cast that spell, then pass the headband to some other character who then gets some benefit from it (whether that be another extra spell, extra skill points, or whatever else). By requiring a 24 hour attunement period, that becomes pretty much impossible. \$\endgroup\$ – Simanos Jul 12 '14 at 15:42
  • \$\begingroup\$ So in effect it's a permanent enhancement bonus, it's not like a permanent raise of the ability like you can get every 4 lvls or by tomes and other means. \$\endgroup\$ – Simanos Jul 12 '14 at 15:43
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Sorry,

Enhancement Bonuses don't stack so it will take the highest value (probably the +4 from Fox's Cunning). Even though the bonus becomes permanent, the type that the bonus is derived from is still the same.

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The spell Fox's Cunning gives a +4 Enhancement bonus to your Intelligence Attribute. This would increase your Intelligence modifier ( And intelligence related skill checks ) by +2.

Also keep in mind when starting to get into the stacking of ability bonuses that you do not try to stack multiple types of Enhancement bonuses together. The Enhancement bonus from Headband of Intellect does not stack with Fox's Cunning. However, the Enhancement bonus from Fox's Cunning would still stack with the Racial Bonus you get for being a Fire elf, the inherent bonus to intelligence from a wish spell would stack with repeat castings of the wish spell up to +5, or the +2 alchemical bonus from Mushroom powder.

A good rule of thumb is that only Dodge Bonuses and Circumstance bonuses stack with themselves. Everything else does not stack. So if you have a +2 to Natural armor as your current race choice equipping an Amulet of Natural Armor +1 will not increase your Armor Class.

Don't forget that only permanent bonuses to intelligence will increase the amount of skill points that you get retroactively. A permanent boost would be a racial bonus, or an inherent bonus. Temporary bonuses such as a Headband of intellect or Fox's Cunning will increase the bonus on skills with a primary intelligence attribute but will not grant retroactive skill points.

When the headband of intellect states that the bonus will become permanent after 24 hours has elapsed, after that amount of time has passed you will still not gain retroactive skill points based on the bonus of the headband. However skill points gained from that point on will be granted if you gain a level.

If you're looking for a real permanent bonus, the Tome of clear thought is going to be your best alternative. However, because you need a wish spell to create them, they are rather pricey.

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    \$\begingroup\$ One thing you have incorrect here: An Amulet of Natural Armor doesn't give a natural armor bonus, it applies an enhancement bonus to your existing armor bonus. In the example you give, the character with a racial +2 Natural Armor bonus would have a total +3 Natural Armor bonus with the amulet. The text says: "... an enhancement bonus to his natural armor from +1 to +5..." \$\endgroup\$ – DuckTapeAl Jul 12 '14 at 5:52
  • \$\begingroup\$ The bonus to Natural Armor is an Enhancement bonus from +1 to +5 depending on the type of amulet. The maximum bonus is +5, the minimum bonus is +1. I'm relatively certain ( ~80% certain ) that an Amulet of natural armor does not stack with existing natural armor. An example of this would be a druid casting barkskin. This amulet and the natural armor enhancement bonus from Barkskin would not stack. \$\endgroup\$ – Sandwich Jul 12 '14 at 5:58
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    \$\begingroup\$ That last part is actually true, because Barkskin also applies an enhancement bonus to your natural armor. However, if you have a bonus to your natural armor from two different sources (say, racial and enhancement), then it stacks just like anything else would. The definition here makes it pretty clear: d20pfsrd.com/basics-ability-scores/… \$\endgroup\$ – DuckTapeAl Jul 12 '14 at 15:20
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The description of the item itself never actually clarifies the effects become permanent after 24 hours, it just says the ability bonus is treated as temporary the first 24 hours. After that, you get the skill bonus points. Both of these seem to still be only effective while your wearing the actual headband. If you were to remove it, you would lose the stat boost, and I am also assuming, the extra skill points. I dont see why this functions any different than a regular headband of intelligence, except for the additional skill points.

All of that being said, the bonus from all 3 items are enhancement bonuses, whether permanent or temporary, the type is the same so they would not stack. This is designed for ability scores on purpose, to avoid really silly high stats by item/spell stacking.

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    \$\begingroup\$ All bonuses to ability scores with a duration longer than 24 hours are treated as permanent after 24 hours. d20pfsrd.com/basics-ability-scores/ability-scores \$\endgroup\$ – Red_Shadow Jul 11 '14 at 18:11
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    \$\begingroup\$ Permanent on the item, not to the character. Maybe i just assumed the intention of the statement in the question wrong. That is what I was referring to, but it may not have been what the OP meant. \$\endgroup\$ – Mayshar Jul 11 '14 at 19:10
  • \$\begingroup\$ The OP says "bonus becomes a permanent bonus as long as it is not removed", I don't think that's his question or a source of confusion to him. \$\endgroup\$ – mxyzplk Jul 12 '14 at 13:27

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