The first bit of advice: Stop trying to "desktop publish" your equations.
Yes Mathematica does display
x^2
and
Superscript[x,2]
almost identically. But try
Solve[x^2==3,x]
and then try
Solve[Superscript[x,2]==3,x]
and see that Mathematica understands how to solve one of those and not the other.
There is also the issue of ' versus \[Prime].
When I un-desktop publish your input, if I haven't made any mistakes, I get
NDSolve[{r^4 Cos[z]^2 f[z]gg[r]-8 f[z]^2 gg[r]-2 r^4 f[z]gg[r]Sin[z]^2-
r^4 Cos[z]gg[r]Sin[z]f'[z]+2 r f[z]^2 gg'[r]== 0,
r^4 Cos[z]^2 f[z]gg[r]^2+4 f[z]^2 gg[r]^2-2 r^4 f[z]gg[r]^2 Sin[z]^2-
r^4 Cos[z] gg[r]^2 Sin[z]f'[z]-r^2 f[z]^2 gg'[r]^2+2 r^2 f[z]^2 gg[r]gg''[r]== 0,
f[0]==f[2 Pi], gg[10]==0.0001,gg'[10]== -0.0001}, {f,gg}, {r,0.01,10},{z,0.0,2 Pi}]
That is closer, but it is still not enough, there is still at least one more problem that I haven't identified.
It seems "unconventional" that f is independent of r and gg is independent of z and that you specify one boundary condition as being equal to another and do not include any derivative in that, but I have not convinced myself whether those are the real problem or not.
