Laplacian[r^k,{r,th},"Polar"]//Simplify
Laplacian[Laplacian[r^k,{r,th},"Polar"],{r,th},"Polar"]//Simplify

Laplacian[r^k,{r,th,f},"Spherical"] // Simplify
Laplacian[Laplacian[r^k,{r,th,f},"Spherical"],{r,th,f},"Spherical"]//FullSimplify

lap[f_,v_,d_]:=D[f,{v,2}]+(d-1)/v D[f,v]

bih=lap[lap[r^k,r,d],r,d] //FullSimplify

biheq=lap[lap[f[r],r,d],r,d] //FullSimplify;

$Assumptions={d>0};

slv=f[r]/.DSolve[{biheq==0,f[1]==1,f'[1]==1},f[r],r][[1]] //Simplify

bihsol=slv/.{C[1]->0,C[2]->0}//Simplify

bihc =Laplacian[Laplacian[x^i y^j z^k,{x,y,z}],{x,y,z}] //Simplify

Limit[bih,{x->0}]