big o notation – What is the complexity of $i^i$?


What is the complexity of the following algorithm in Big O:

for(int i = 2; i < n; i = i^i)
{
    ...do somthing
}

I’m not sure if there is a valid operator to this type of complexity.
My initial thought was as follows:

After $k$ iterations we want: (using tetration?)

${^{k}i} = n implies k=logloglog…_klog{n}impliesmathcal{O(logloglog…_klog{n})}$ (where we have k times the log function) but i’m not sure if this is evan a valid way of writing this.
Anyway, we have a complexity that that includes $k$, which does not seems right to me.