Just to throw something in, I computed the image of the Cartesian grid for the matrix power super exponential. This can directly be compared with

Figure 7 in D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathermatics of Computation, in press (200, and it looks quite similar:

The picture shows the image of lines with constant real part. The thick red lines are -1,0,1,2.

Figure 7 in D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathermatics of Computation, in press (200, and it looks quite similar:

The picture shows the image of lines with constant real part. The thick red lines are -1,0,1,2.