The following provides the Schroeder and Abel functions in Mathematica using the support for solving recursive equations and Bell polynomials. The code takes about a hour to run for the first ten derivatives.
Code:
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}]
Daniel