How do Unseen Seer and similar prestige classes affect the spell-casting progression of a character with Assassin levels?

Question

Assassin is prestige class from DMG with 10 levels and arcane spells progression. However, it does not progress arcane spells of existing class, but it has its own spell list and spells progression in contrast to some other prestige classes which continue progress on existing arcane casting class (via +1 level of existing arcane spellcasting class), like Unseen Seer.

Unseen Seer from CM also has 10 levels, and on each level progresses existing arcane spellcasting class.

What happens when character is already level 10 Assassin and takes level in Unseen Seer (or other prestige class that continues spell progress in such way)?

There is not Assassin's spells progression beyond level 10.

For level 10 Assassin, do levels of prestige classes, which advance spells progression, only increase character's caster level?

Answer 1

The answer to your first question can be found here, including lots of details about how we arrive at this conclusion:

Yes,

assassin is a spellcasting class that can be chosen by other prestige classes that advance spellcasting.

In effect, the test that is near-universally agreed-upon as being most fair, sensible, and consistent with the rules (which are, unfortunately, ambiguous) is this:

A class that advances spellcasting is not, itself, a spellcasting class; a spellcasting class is one that has its own spells.

So unseen seer can advance an assassin’s spellcasting.

As for what happens if, say, you are a Rogue 5/Assassin 10/Unseen Seer 1 and you choose assassin as your class, that is very contentious. We actually have a separate answer on that here. Again, the rules are ambiguous, but I suggest that the most reasonable approach is to advance only caster level in these cases (though I might houserule something for an assassin here, because they could use the boost, I just would be clear that it was a special houserule for the assassin and not precedent to do the same with the vastly-more-powerful ur-priest).