User blog:Primussupremus/Is it possible to have pi up arrows in Knuth up arrow notation?

I was just wondering if it is possible to have pi{pi up arrows}pi? in other words is this a valid expression or are knuth up arrows only valid for positive integers?

A further question I have is on whether or not real numbers can be used as up arrow values for example can we have phi^pi up arrows?

Going beyond that is it possible to have complex numbered up arrows?