Lets define function T as $$T(0) = \aleph_0$$ $$T(1) = \aleph_{\aleph_0}$$ $$T(2) = \aleph_{\aleph_{\aleph_0}}$$ etc

No finite tower of alephs can reach the first inaccessible cardinal

My questions are:

1. Can we 'feed' infinite ordinal numbers as a parameter to function T? I read somewhere - it was about the inaccessible cardinals - that the size of the tower is countable, but is it just a limitation of the first order ZFC with finitely-sized formulas? Can generalized function T() be expressed in, say, second order ZFC?

2. So if there is nothing wrong with that definition of T, what is relative size of, say, $$T(\omega+1)$$ $$T(\omega^2)$$ $$T(2^{\aleph_0})$$ Are they still below the first inaccessible cardinal?

3. Obviously, it leads to the last question: How far can we go with it, where is the fixed point of T relative to the hierarchy of the cardinals?

Thank you