User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180205184553

"Answer why Taranovsky's notation is so powerful."

I can tell you right away if you want to.

"Extend Taranovsky's notation to transfinite-indexed cardinals and beyond."

You mean things like $$C(\Omega_\alpha,0)$$? Not happening. TON is very difficult to define for transfinite $$\alpha$$. It's usually either impossible to make it work, or the resulting notation doesn't go much further than regular TON. This has to do with the very nature of TON and one of it's main properties called "n-shiftedness". This is also connected to the answer for why it's so strong.