Definition of the function $f_{\epsilon_0}$ in the fast-growing hierachy I found an article where the growth of 
$$f_{\epsilon_0}$$
in the fast-growing-hierachy is described, but the text 
 is very long and difficult to understand.
Is
$$f_{\epsilon_0}(n)=f_{\omega^{\omega^...}}(n)$$ with n omegas 
or something else ?
I think I understood the procedure upto
$$f_{\omega+1}(n)$$
but I already have difficulties to understand
$$f_{\omega^\omega}(n)$$
Can someone explain the procedure step by step, beginning with
$$f_{\omega+1}(n)\ ?$$ 
 A: Yes. The standard way of defining these sequences goes by assigning in an explicit fashion to each limit ordinal $\alpha$, for as long as possible, an increasing sequence $\alpha_n$ that converges to $\alpha$. Once this is done, we can define $f_\alpha$ by diagonalizing, so $f_\alpha(n)=f_{\alpha_n}(n)$ for all $n$.
Of course there are many possible choices of sequences, but one can argue that the standard choices are the natural ones. Here, we note first that if $\alpha<\epsilon_0$, then we can write $\alpha$ uniquely as $\omega^\beta(\gamma+1)$, for some $\beta<\alpha$. The ordinal $\alpha$ is limit iff $\beta>0$. In that case, we set $\alpha_n=\omega^\beta\gamma+\omega^{\beta_n}$, if $\beta$ itself is a limit, while if $\beta=\delta+1$ is a successor, then we set $\alpha_n=\omega^\beta\gamma+\omega^\delta n$. 
The approach described above does not work when $\alpha=\epsilon_0$, since we would have ($\gamma=0$ and) $\beta=\alpha$. The natural choice here is to let $\alpha_0=0$ and $\alpha_{n+1}=\omega^{\alpha_n}$, so $\alpha_{n+1}$ is a tower of $\omega$s of length $n$. 
With this convention in place, we can continue for a while. Eventually we reach again a point where we need yet another convention. The topic of ordinal notations studies ways of continuing the process for as long as possible. The goal is to be able to refer explicitly to (recursive) ordinals in inductive arguments, and the notations quickly become so complicated that we need stronger and stronger background theories to verify that the orderings they capture are in fact well-orderings.

The sequence of functions $f_\alpha$, defined by Löb and Wainer, sets $f_0(n)=n+1$, and  $f_{\alpha+1}(n)=f_\alpha^n(n)$, where the superindex denotes iteration, that is, $f_\alpha\circ\dots\circ f_\alpha(n)$, where $f_\alpha$ appears $n$ times. Finally, if $\alpha$ is limit, then we set $f_\alpha(n)=f_{\alpha_n}(n)$, as indicated above. As long as the choice of cofinal sequences is reasonable enough, we can verify that the resulting transfinite sequence of functions is strictly increasing, where for functions $f,g:\mathbb N\to\mathbb N$, we set $f<g$ iff $f(n)<g(n)$ for all but finitely many values of $n$. 
For as long as we have been able to define these functions (which means, for as long as our system of ordinal notations allows us), the $f_\alpha$ are recursive. The computation of the values $f_\alpha(n)$ depends directly on the complexity of the ordinal notation we use. 
(This explains the choice of $\alpha_0=0$ rather than $1$ or $\omega$ for $\alpha=\epsilon_0$. This choice is picked so that the recursive functions provably total in the theory called $I\Sigma_{k+1}$ are eventually dominated by $f_{\alpha_{k+1}}$ for all $k$. Here, $I\Sigma_{k+1}$ is the sub-theory of first-order Peano arithmetic resulting from restricting the induction schema to $\Sigma_{k+1}$ formulas.)
For example, $f_\omega(n)=f_n(n)$ and $f_{\omega+1}(n)=f_\omega^n(n)$. For example, $f_{\omega+1}(0)=0$, $f_{\omega+1}(1)=f_\omega(1)=f_1(1)=f_0^1(1)=f_0(1)=2$, $f_{\omega+1}(2)=f_\omega(f_\omega(2))=f_\omega(f_2(2))=f_\omega(f_1(f_1(2)))=f_\omega(f_1(f_0(f_0(2))))=f_\omega(f_1(f_0(3)))$ $=f_\omega(f_1(4))=f_\omega(f_0^4(4))=f_\omega(8)=f_8(8)$, etc. This is already a significantly large number. 
To compute $f_{\omega^\omega}(n)$, we would start by noting $f_{\omega^\omega}(n)=f_{\omega^n}(n)$, so $f_{\omega^\omega}(0)=f_1(0)=0$, $f_{\omega^\omega}(2)=f_{\omega^2}(2)=f_{\omega2}(2)=f_{\omega+2}(2)=f_{\omega+1}(f_{\omega+1}(2))=f_{\omega+1}(f_8(8))$, etc. In practice, the computation of these functions is quite unfeasible. 
For a last example of how hopeless this quickly gets, let's attempt to see how $f_{\omega^\omega}(3)$ would be computed. Let's write $\alpha\to\beta$, for $\alpha$ limit, to indicate that $\beta=\alpha_3$, so that $f_\alpha(3)=f_{\alpha_3}(3)$.
We have $\omega^\omega\to\omega^3\to\omega^23\to\omega^22+\omega3\to\omega^22+\omega2+3$, so that $f_{\omega^\omega}(3)=f_{\omega^22+\omega2+3}(3)$. In turn, to compute this we must compute $f_{\omega^22+\omega2+2}^3(3)$, which in particular leads us to compute $f_{\omega^22+\omega2+2}(3)$, for which we then need to compute $f_{\omega^22+\omega2+1}^3(3)$, which eventually leads us to compute $f_{\omega^22+\omega+3}(3)$, etc. 
Similar computations for larger ordinals are even more cumbersome. For instance, with the same convention, we have
 $$\omega^{\omega^\omega}\to\omega^{\omega^3}\to\omega^{\omega^23}\to\omega^{\omega^22+\omega3}\to\dots\to\omega^{\omega^22+\omega2+2}2+\omega^{\omega^22+\omega2+1}2+\omega^{\omega^22+\omega2}2+\omega^{\omega^22+\omega+2}2+\omega^{\omega^22+\omega+1}2+\omega^{\omega^22+\omega}2+\omega^{\omega^22+3}\to\dots$$
One can check by induction that $f_1(n)=2n$, $f_2(n)=n2^n$, and $f_3(n)$ is already larger than a stack of powers of $2$ of length $n$. (Some authors define the sequence slightly different to ensure that $f_2(n)=2^n$ and $f_3(n)$ is precisely the stack of powers of $2$ of length $n$.) These functions form what we usually call a fast-growing hierarchy.
A while ago, I wrote a paper on Goodstein's function that required me to consider this sequence. You may find it here.
