My question would be, just how high can you get before you miss a whole number?
Infinity (or whatever arbitrary limit of single-arity operations might be applied). I know it's considered gauche around here to read the source article, much less a video, but it gives the formula and process which allows any integer to be reached.
With "sqrt()" being the square root function:
The log base sqrt(4)/4 of [log base 4 of sqrt(4)] = 1.
The log base sqrt(4)/4 of [log base 4 of sqrt(sqrt(4))] = 2.
The log base sqrt(4)/4 of [log base 4 of sqrt(sqrt(sqrt(4)))] = 3.
The number of times the square root function has been applied in the inner logarithm, is the integer which results from the formula. Therefore, you can create any positive whole number with four fours (and an indefinite number of operations).
You can bring any calculator you like to the midterm, as long as it doesn't dim the lights when you turn it on. -- Hepler, Systems Design 182