I created http://tetration.org which explores what lies beyond exponentiation. My life's work has been to extend the Ackermann function to the complex numbers and matrices. I believe my website lead to a renaissance in tetration research. The following Mathematica code replaces a thousand lines of earlier code. It computes the flows from maps. Combined with the historic three argument Ackermann function it allows the Ackermann function to the extended to the complex numbers.

order=10;

H[0]=0;

H[1]=f'[0]^t ;

Do[H[max]=First[r[t]/.RSolve[{r[0]==0,r[t]==Sum[Derivative[k][f][0]BellY[max,k,Table[H[j]/.t->t-1,{j,max}]],{k,2,max}]+ f'[0] r[t-1]},r[t],t]],{max,2,order}];

Schroeder=f'[0]^t z+Sum[1/k! H[k]z^k,{k,2,order}]

Abel=Limit[Schroeder,{f'[0]->1}]