I'm still trying to figure out where the +3 caster levels come from to get a 7th level caster up to a 10d6 fireball. And that is my greatest complaint with this guide: statements are made that aren't fully explained so that someone who needs a guide like this one won't find it as useful as they might.
That's a fair complaint, I'll update the guide later.
But however, this is not a guide about how to make a 7th level character that can beat damage records. I gave that as an example of what is possible while in the realm of "blasting", to refute the common argument that "blasting is underpowered". I do not advice to build such a "one poney trick" character.
That said, +3 levels could come from:
1) being Gnome Pyromaniac alternate racial trait (advanced pathfinder guide) gives +1 Caster Level to Fire spells.
2) Varisian Tatto Feat (which gives you +1 Caster level to Evocation Spells)
3) Loreseeker Background trait, which allows you to select 3 spells from your spell list, and cast them at +1 caster level.
This is just one way to get it. You could go other routes. For example:
Bloatmage Initiated feat gives you a +1 to a school of magic (evocation in this case) too, but limits your max dexterity and slows you
Gifted adept background trait gives also +1 caster level to a single spell of your choice, although it's hard to (note, you can get 2 extra background feats if you need it).
Orange Ioun Stones also give +1 CL.
Spell Specialization feat gives you +2 to one single spell.
Spell perfection double any bonus given to a spell by a feat (so you could double the CL bonuses), but is a high level feat with a strong requisite, just for completeness.
This is just a few ways to get to +3. You could mix and match some of those, if you want, or use them all together to get +8 CL (although not possible at 7th).
That said, once I've re-checked, I think the problematic paragraph was refering to *7th level spells* and *14th level caster*.
So it should read:
a 14th level caster can cast 7th level spells. That means 14d6+14 with a delayed fireball (and draconic blood), or 5d6+75 if you empower and maximize a 3rd level fireball. In adition, a 3rd level fireball raised to level 7 spell slot -using magical linage-, is a 3rd level spell regardless. So it bounces in Globes of invulnaribility, but also work with Lesser Quickened MEtamagic Rods. Casting two 5d6+75 spells in one round, for 10d6+150 (average 185) surely beats casting a regular delayed fireball for 14d6+14.
