Threefold aspect, extend spell, ???, profit!

The spell threefold aspect from the APG has a duration of 24h, and can grant a +4 bonus to Intelligence. With extend spell, the duration is 48h.

According to the glossary in the CRB, "ability bonuses with a duration greater than 1 day actually increase the relevant ability score after 24 hours. Modify all skills and statistics related to that ability. This might cause you to gain skill points, hit points, and other bonuses. These bonuses should be noted separately in case they are removed."

So, does an extended threefold aspect grants me additional skill points and spell slots? Does it allow me to change my skills every two days?

Seems slightly cheesy but legal. It won't stack with a headband of intellect though, so I don't think there's really a problem. You'll only be able to pull it off at level 9 max anyway, or level 7 if you take magical lineage for it (don't do that, you'll regret it).

He has a point though, particularly in that every day or two he can be fully trained in two different new skills.

At level 9 he can get +9-12 to two skills of his choice. Eh, there are better uses for spell slots. Consider invisibility gives +20 to stealth while moving and you can do that at level 5.

Not only that, but according to the wording of permanent bonuses, you would need to prepare a full day in advance to get this skill bonus. Nothing on the fly here, not overpowered.

That seems like a great way to do crafting.

If you let the spell expire and re-cast to switch which skills are given bonuses then the Intelligence boost is new (it has no longer lasted one day or longer). So you'll need to wait a second day to "turn on" the new skill boost selection.

I have a Witch that keeps Threefold Aspect on all the time using an Extend rod. By casting it an hour earlier each day it is effectively permanent (so far nothing has dispelled it). Every now and then she must cast it on two consecutive days so it does not expire while she is sleeping.

Is it worth losing a 4th level spell slot to reallocate the money the PC would spend on a Intelligence boosting headband? Perhaps. That depends upon party and player and campaign.