User blog comment:Simplicityaboveall/Extremely Large Numbers 3/@comment-29014792-20160801191948/@comment-5529393-20160801212100

It is impossible for any notation system to have a limit of w_1, as there are uncountably many ordinals below w_1 but only countably many notations in any notation system.

Since heshbar-q is defined in a computable fashion (unless there is something uncomputable in the definition of the R function, when you do get around to defining it), heshbar-q will not reach w_1^CK either. The limit will be some computable ordinal - basically, how far the "complete hereditary form to base w" reaches.