Basic Examples 
(3)
 

BinaryIteratedLog
is the inverse of binary tetration (repeated exponentiation):
In[61]:=
[◼]
BinaryIteratedLog
[2^2^2^2]
Out[61]=
4
———
A slightly larger input shows a step-like jump in the value of
BinaryIteratedLog
:
In[63]:=
[◼]
BinaryIteratedLog
[2^2^2^2+1]
Out[63]=
5
———
Make a table of the binary iterated logarithm of the first 30 integers:
In[65]:=
Table
[◼]
BinaryIteratedLog
[i],{i,0,30}
Out[65]=
{0,0,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4}