How to compute the infinite tower of the complex number $i$, that is$ ^{\infty}i$

In case you really mean $ I^x - x = 0 $ you can find this by the iteration

 x = <some initial value> 
 repeat
     x = I^x   // where I is the imaginary unit
   until convergence

You'll get approximately $ x = 0.438282936727 + 0.360592471871*I $ (using Pari/GP, for instance)

However, you can also find the value using the lambert-w-function.
[update] using the lambert-w:

let $ \lambda=\ln(i) $
then
$i^x = x$
$1 = x* \exp( -x \lambda) $
$-\lambda = -x \lambda * \exp( -x \lambda) $
$ W(-\lambda) = - x \lambda $
so
$ x = \frac{W(-\ln(i))}{-\ln(i)} $

(Whether this is more "exact" is rather a question "exact in terms-of-what?")

[end update]