User blog:Ubersketch/Extended Ackermann Function

I noticed something really cool when I analyzed the ackermann function in the fast iteration hierarchy. Basically, a+{[2]b[1]c} is the ackermann function. But what about a+{[3]b[2]c[1]d}? Or beyond? Does anybody have any idea of any notation or function which is identical to this extended ackermann function?