The lowest large cardinals can be seen as a simple recursive rule that defines an unbounded sequence, then defines a fixed point of the sequence, then an unbounded sequence of fixed points, a fixed point of this new sequence, an new unbounded sequence, and so on recursively (I am not sure if the word recursive is well employed in this context). And if I am not wrong, the starting sequence of large cardinals, such as inaccesible, 1-inaccessible, α-inaccessible, hyperinaccessible, α-hyperinaccessible, hyperhyperinaccessible, hyper$^α$-inaccessible, Mahlo, 1-Mahlo, α-Mahlo, hyper-Mahlo and α-hyper-Mahlo more or less (or exactly?) follows this pattern. However, my understanding is that the definition of the next stage (or close to next) large cardinal, the weakly compact cardinals, breaks the pattern, because it is not defined in terms of fixed points. I think that the same happens with the rest (or most of them) of the large cardinals defined up to now. The larger ones being defined in terms of nontrivial elementary embeddings (or something like that that I still fail to understand).
The recursive definition that I mentioned at the beginning and that reaches more or less up to the α-hyper-Mahlos is used only a few times and with no consistent computable ordinal notations.
So my questions are two related ones: 1) Is there a consistent ordinal notation system to reach as high as possible in the recursive hierarchy of taking limits of unbounded sequences? 2) If there is one, how high do we get, arbitrarily high? above or below the largest known cardinal? Can all of the large cardinals defined up to now be alternatively defined in this way too?
I apologize if there are two many questions, I can try to write them separately.