A computation: the Mathematica 8 commands:

```
F[A_]:=F[A]=-((A ProductLog[-1,-(Log[2]/A)])/Log[2]);
N[NestList[F, 2, 15]]
```

gives the following breakpoints for $T$:

$2.,4.,16.,108.099,1090.86,15148.6,273611.,6.17193\times 10^6,$

$1.68677\times 10^8,5.45588\times 10^9,2.05015\times 10^{11},8.81634\times 10^{12},$

$4.2853\times 10^{14}, 2.32997\times 10^{16},1.40462\times 10^{18},$

$9.31777\times 10^{19}$

That is, $T(n)=1$ for $2 < n \leq 4$, $T(n)=2$ for $4 < n \leq 16$, $T(n)=3$ for $16 < n < 108.09$, and so on.

This suggests that $\log_2 n \log_2\log_2 n$ is significantly too fast. Perhaps the square root of this is closer to the truth.