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Say you're a 14th-level Sorcerer with 1 level in Archmage who then goes Fighter. You take Practiced Spellcaster (CA p.82) to boost your Sorcerer CL, and then later take it again to boost your Archmage CL.
Since Archmage inherits its spellcasting from Sorcerer, does this mean that at character level 20th, your caster level for everything is 14+4+1+4=20, so even though you don't know 8th/9th-level spells, your Delayed Blast Fireball deals its full 20d6 (for example)?
No.
Counterintuitive as it may seem, Archmage is not a spellcasting class. Instead, it advances another class’s spellcasting. So when practiced spellcaster says “pick a spellcasting class,” Archmage is not a valid choice.
You can see this discussed often online when someone thinks of applying Archmage or similar prestige class’s spellcasting advancements to mystic theurge (which is similarly not actually a spellcasting class).
WotC could have avoided much confusion had they drawn a clearer line on what classes actually were spellcasting classes. However, as a strong general rule, spellcasting classes have a class feature named “spells,” and the non-spellcasting classes have a class feature named “Spells per Day/Spells Known.” A few books instead use “Spellcasting” for both, which is less helpful, but the same approach applies: only classes that actually give you spells are spellcasting classes, the classes which advance other classes’ spellcasting are not.
Archmage doesn't have spellcasting but rather extends the base class' spellcasting, as described in +1 level of existing arcane spellcasting class.
Therefore you can't take Practiced Spellcaster for Archmage or any other class that doesn't have its own spellcasting progression.
