Q:

Calculation of $f_{\omega^3}(2)$ in the fast growing hierarchy

How is the number 
$$\large f_{\omega^3}(2)$$
in the fast growing hierarchy calculated ?
My only idea is to convert to 
$$\large f_{\omega^2 2}(2)$$
but now I have no idea how to continue.

A:

Too large to calculate!
I assume you are referring to the definition on Wikipedia: http://en.wikipedia.org/wiki/Fast-growing_hierarchy
If so, then, as the article points out, $f_{\omega + 1}(64)$ is already larger than Graham's number. $f_{\omega^3}$ is much larger than this.
But perhaps you're asking what an algorithm could be if we have unbounded time and space? Well here's a start:
$f_{\omega^3}(2) = f_{\omega^2 \cdot 2}(2) = f_{\omega^2 + \omega^2}(2) = f_{\omega^2 + \omega \cdot 2}(2) = f_{\omega^2 + \omega + \omega}(2) = f_{\omega^2 + \omega + 2}(2) = f_{\omega^2 + \omega + 1}^2(2) = f_{\omega^2 + \omega + 1}(f_{\omega^2 + \omega + 1}(2)) = f_{\omega^2 + \omega + 1}(f_{\omega^2 + \omega}^2(2)) = f_{\omega^2 + \omega +1}(f_{\omega^2 + \omega}(f_{\omega^2 + \omega}(2))) = f_{\omega^2 + \omega +1}(f_{\omega^2 + \omega}(f_{\omega^2 + 2}(2))) = f_{\omega^2 + \omega +1}(f_{\omega^2 + \omega}(f_{\omega^2 + 1}(f_{\omega^2 + 1}(2)))) = f_{\omega^2 + \omega +1}(f_{\omega^2 + \omega}(f_{\omega^2 + 1}(f_{\omega^2}(f_{\omega^2}(2))))) = f_{\omega^2 + \omega +1}(f_{\omega^2 + \omega}(f_{\omega^2 + 1}(f_{\omega^2}(f_{\omega + \omega}(2))))) = f_{\omega^2 + \omega +1}(f_{\omega^2 + \omega}(f_{\omega^2 + 1}(f_{\omega^2}(f_{\omega + 2}(2))))) = f_{\omega^2 + \omega +1}(f_{\omega^2 + \omega}(f_{\omega^2 + 1}(f_{\omega^2}(f_{\omega + 1}(f_{\omega + 1}(2)))))) = f_{\omega^2 + \omega +1}(f_{\omega^2 + \omega}(f_{\omega^2 + 1}(f_{\omega^2}(f_{\omega + 1}(f_{\omega}(f_{\omega}(2))))))) = f_{\omega^2 + \omega +1}(f_{\omega^2 + \omega}(f_{\omega^2 + 1}(f_{\omega^2}(f_{\omega + 1}(f_{\omega}(8)))))) = f_{\omega^2 + \omega +1}(f_{\omega^2 + \omega}(f_{\omega^2 + 1}(f_{\omega^2}(f_{\omega + 1}(f_8(8)))))) = ...$
At this point I get bored of writing this out by hand, but you get the point, you just keep applying the rules. At some point the numbers will get too large to write down.
Why do you ask about calculating this number?

