A character has a base INT score of 20 and dons a headband +6. A week later, his INT is now 26.
At INT 20, his spell level maximum was up to 5th level spells.
Does he now have access to 8th level spells per the INT of 26?
I think you're confused.
To case 0 level spells you need a casting stat of 10.
To cast 1st level spells you need a casting stat of 11
To cast 2nd level spells you need a casting stat of 12
Etc.
So if you have a casting stat (int in this case) of 19 you can cast all available spell levels.
You correct, I am dumb. I wasn't looking at the wizard spells per day but the BONUS spells per day. Been playing too long today!
Lol. No worries. It happens to us all. Good gaming.
Even though the example was incorrect, the question is still valid. Let's instead pretend the wizard has a base Intelligence of 15, which normally allows him to cast 5th level spells. If he puts on a headband of intellect +4, he will now instead be able to cast 9th level spells, assuming he's 17th level and knows any (which he probably doesn't since he couldn't cast them until he got the headband).
He will also gain bonus spells of 3rd and 4th level, as appropriate for his newly enhanced Intelligence, and his spells will get +2 save DC.
Note that even though our relatively dim archmage can only cast 5th level spells with Int 15, he will still get spell slots of higher levels. He can use these to prepare lower-level spells, possibly boosted with metamagic feats (other than Heighten Spell).
While he might not get bonus spells in the first 24 hours due to the "temporary" nature of the increase, he can get the increased DC and be able to cast higher spells immediately. Probably from a scroll.
He can have spells higher than he can cast in his spellbook. Just need sufficient spellcraft to add them. He can also have slots for high level spells without using high level spells in them. For example, if he has a 9th level slot, but a 5th level casting cap, he can store a 5th level spell in the 9th level slot. He can also use potions and spells to pump his stat, and while boosted, use the slot with a higher level spell. Filling the slot with a high level spell might take a bit of work before the potion/spell wore off.
/cevah