User blog comment:PsiCubed2/My Letter Notation/@comment-80.98.179.160-20171107122831/@comment-30754445-20171107204345

This idea was actually considered and dropped in Part II:

"It would be natural to continue by defining I as repeated H and J as repeated I and so on, but then we'll reach the end with Zn being equivalent to a puny 10↑22n and we obviously don't want that."

So the final definition of J actually grows much much faster than what you've suggested. In fact, your "Z10" is exactly equal to J22.