Back in high school I wondered if there is a such a function that has higher growth rate than exponential function. Several years later it came to my mind that this is obvious because we can create arithmetic operators. I'm not a mathematics graduate so it was big thing to me - I thought we are rather bounded to these 5 operators "$+ -\times /$ ^" and we can't do anything about it.
Example:
$2+2+2+2=2\times4=8$
so:
$2\times2\times2\times2=2$^$4=16$
so:
$2$^$2$^$2$^$2$$=2☮4=256$
so:
...
and so on.
So we can use this new "☮" operator instead of "^" to get a function with more steeper increment rate. What is a formal name for such a ☮ operator, what do we use it for? (Some time ago I've found out that Donald Knuth applied such iterated exponentiation only to notate large integers. I see we can use it to count bacteria population, given generation number). Why is such an operator not more common used?
After I "discovered" that ☮ operator I realized that there are also new operators at levels below addition:
$2+2+2+2=2\times 4=8$
because:
$2222=2+4=6$
because:
$...............=24=5$ (According to my calculations it equals "5")
because:
........
And this operator really blew my mind. It so bizarre and exotic to me. The same questions: what is the formal name, application and properties for that operator (and those level below it)?