2.86295 + 3.22327 i

[Gottfried]

Let z^z = z.

Take ln on both sides.

z ln(z) = ln(z) + 2 k pi I 

Now if we let k be 1 we get

2.86295 + 3.22327 i = z

And if k = -1 we get

2.86295 - 3.22327 i

And they appear the smallest ones.
That is probably easy to prove though I did not.

Yes, your ansatz is similar to my first ansatz in my linked MSE-problem asking for roots z^z^z+1 which I also rewrote initially in its logarithmic form, and did not consider the multiplicity of complex logarithms.      (see https://math.stackexchange.com/a/1415538/1714)                       

Also, the form \( f^{\circ2}(1)=0 \) directs immediately to the conjugate expression for z^z (and I''ve indeed cancelled the z-factor (which introduces the need to handle z=0 extra!) I would have liked to work directly with this, say an adapted LambertW-function or so, but hadn't a good idea.                     

I'm curious whether we find more zeros and besides the interpolation-line indicated by my 14 solutions...

update: I'm now -after the newest image in my older answer- convinced there are no further roots besides that of the rough line shown in my previous post.