log*n is called the iterated logarithm of n, defined as the count of how many times you apply the logarithm function iteratively such that the value is less than or equal to 1.
eg, log*16=3
The recurrence relation is as follows :
log*n = 0 , if n<=1
= 1 + log*(log n) , if n>1
log 16 = 1+log*(log16) = 1+log*4
= 1+1+log*(log4) = 1+1+log*2
= 1+1+1+log*(log2) = 1+1+1+log*1 = 1+1+1+0 = 3
Notice that the iterated logarithm function grows at an extremely slower rate compared to logarithm function.
For a very large value of n, log*n will be a very small value compared to logn.
eg, for n=1024, log n=10 and log*n=4
n= 265536 , log n = 65536 and log*n=5
Also, log*n will take one less iteration on log*(logn) [ Look at the recurrence relation ]
log*n -1 = log*(logn)
Consider log*n=x,
log(log*n) = log x
log*(log n) = log*n - 1 = x - 1
clearly, log*(log n) is asymptotically larger than log(log*n)