restart;

f := z -> exp(exp(z));

f(x+I*y);

assume(x,real); assume(y,real);

u := Re(f(x+I*y)); v := Im(f(x+I*y));

uxx := simplify(diff(u,x,x));

uyy := simplify(diff(u,y,y));

simplify(uxx+uyy);

diff(f(z),z);