This is not a answer.

It's just a helper to discuss some things about the question, because is too large for the comments.

Looks like $-\sqrt{2}$ isn't a solution for the equation, but I'm not sure. Looks like too, the power tower of a number should converge only on a specific interval ($[e^{−e},e^{1/e}]$).

But using Mathematica and the ProductLog function (wich the Lambert $W(z)$ function) we find some strange things:

Using $h(z)=z^{z^{z^{\ldots}}}=-\frac{W(-\log (z))}{\log (z)}$ (`h[z_]:=(-ProductLog[-Log[z]])/Log[z]`

)

Calculating the power tower to $\sqrt{2}$ we have
`N[h[Sqrt[2]], 10]=2.000000000`

And the power tower to $-\sqrt{2}$ we have
`N[h[-Sqrt[2]], 10]=0.2513502988 + 0.3162499180 I`

Calculating explicity, by iteration

${-\sqrt{2}},{(-\sqrt{2})}^{({-\sqrt{2}})},{(-\sqrt{2})}^{({-\sqrt{2})}^{\ldots}}$ we have

```
Table[N[Re[PowerTower[-Sqrt[2], i]], 30] +
I*N[Im[PowerTower[-Sqrt[2], i]], 5], {i, 1, 15}] // TableForm
```

```
-1.41421356237309504880168872421
-0.163093997943414854921937604558+0.59044 I
0.140921295793052749536215801866-0.044791 I
1.10008630700672531426983704055+0.50079 I
-0.268168781568546776692908102136-0.14235 I
0.894980750563013739735614892750-1.1090 I
-33.5835630157562847787187418023+29.118 I
6.49187847255812829134661655850*10^-46-1.5181*10^-45 I
1.00000000000000000000000000000+1.5134*10^-45 I
-1.41421356237309504880168872421-2.2930*10^-44 I
-0.163093997943414854921937604558+0.59044 I
0.140921295793052749536215801866-0.044791 I
1.10008630700672531426983704055+0.50079 I
-0.268168781568546776692908102136-0.14235 I
0.894980750563013739735614892750-1.1090 I
```

Ploting the real and imaginary part of the function $h$, we have:

To the real part:

```
Plot[Re[N[h[x]], {x, -2, 0},
Epilog -> {PointSize[0.01],
Point[{-Sqrt[2],
N[Re[h[-Sqrt[2]]]]}]}]
```

and to the imaginary part:

```
Plot[Im[N[h[x]], {x, -2, 0},
Epilog -> {PointSize[0.01],
Point[{-Sqrt[2],
N[Im[h[-Sqrt[2]]]]}]}]
```

So looks like the function converges, but, unfortunally not to $2$.

I will post this for now, but, maybe I will create a new question just to treat this convergence and I will embrace a answer from here.

Please if someone can clarify this a bit, left a comment.

Thx.