Determining effect of powers of DFT matrix conjugate on input without multiplying
In both cases (odd and even powers of 2) you should get as a result simply a time inversion of your original signal. Because of the implicit periodicity you get from the original vector

$$[x_1,x_2,x_3,\ldots,x_N]$$

the following time-inverted (and periodically continued) vector

$$[x_1,x_N,x_{N-1},\ldots,x_2]$$

Here's an 'odd power' example:

N=8;F=dftmtx(N);U=F/sqrt(N);
x=[1 2 3 4 5 6 7 8]';
y=U'*U'*x;
y =

1.00000 - 0.00000i
8.00000 + 0.00000i
7.00000 + 0.00000i
6.00000 - 0.00000i
5.00000 + 0.00000i
4.00000 + 0.00000i
3.00000 + 0.00000i
2.00000 - 0.00000i