Big numbers

We once learned to count, and for a while, 10 seemed like a big number. Then 100 seemed the limit of numbers, until one thousand came...

As we learn how to count without a limit (you do know what forty-five thousand, six hundred and seventy-eight plus one is right...), it seems that we have mastered all ways of expressing numbers, no matter how big. Yet, this is often nearly not the case in practice. When writing out the rest mass of an electron, we do not write 0.000000000000000000000000000000911 kg. (Well usually we don't, don't be a weird person)

Instead, we use a more powerful, space saving notation for numbers of this small magnitude: 9.11*10^-31. Neat. This also works for larger numbers, like 3.14*10^8=314000000. Looks perfect, but is not quite the best yet.

This is also clearly a use (abuse) of the power notation, where a^b= a*a*a*a*a*a... b times. So how about we get a little creative.

Note:
a*b = a+a+a+a+a+a+a+a... b times
a^b = a*a*a*a*a*a*a*a... b times

We shall define tetration as a b, where the power is applied b times. And we notice that we can extend it further. Hmm... Conway chain notation... ...

Too mind-bogglingly big, and suspect usefulness. Ah well. But anyway, a rather skimpy list of functions in order of bigginess:

+ < */polynomial < ^ < ! < tetration