What does Frattini length measure? I have heard derived length, for example, described as a measure of "how non-commutative" the group is.  An abelian group will have derived length $1$, whereas a non-solvable group will be so non-commutative that the even the fastest converging commutator series will continue infinitely.
Analogously, nilpotency class measures "how close to being abelian" a group is by finding the length of the slowest converging commutator series.  A group of nilpotency class $2$ has essentially only one "twist" in its structure, and a group sufficiently far from being abelian cannot be completely unraveled by commutators without using bigger guns.
Then, between these two definitions, we've got Fitting length, which measures how far a group is from being nilpotent.
Of course, there is another series floating around out there.

The Frattini subgroup of a group $G$, denoted $\Phi(G)$, is the intersection of all maximal subgroups of $G$.  (Equivalently, $\Phi(G)$ is the set of all non-generators of $G$.)  We define the Frattini series by $\Phi_0(G)=G$ and $\Phi_{k+1}(G)=\Phi(\Phi_k(G))$.  The smallest $n$ for which $\Phi_n(G)=1$ is the Frattini length of $G$.

So, intuitively, what does Frattini length measure?
 A: A chief factor of a finite group $G$ is a pair $H/K$ such that $K \leq H$ are two normal subgroups of $G$ such that there is no other normal subgroup of $G$ properly contained between $H$ and $K$. Chief factors of finite groups come in two major flavors: elementary abelian $p$-groups and a direct product of copies of a fixed finite non-abelian simple group.
There are three kinds of chief factors: complemented, Frattini, and the rest. A complemented chief factor is one in which there is some $M/K$ such that $G=MH$ and $M \cap H = K$. A Frattini chief factor is one where $H/K \leq \Phi(G)/K$. A complemented factor can be abelian or not, but a Frattini factor must be abelian. The rest consist only of non-abelian chief factors, and I don't know of any examples offhand (they'd have to be pretty big).
If $H/K$ is a chief factor and $H \leq \Phi(G)$, then of cours $K \leq \Phi(G)$ and so $\Phi(G/K) = \Phi(G)/K$ and $H$ is a Frattini factor. Conversely, a chain of Frattini factors $H_1/H_2, H_2/H_3, \dots, H_n/1$ will all live in the Frattinis subgroup: $H_1 \leq \Phi(G)$. So the Frattini subgroup is the bottom-most section of Frattini chief factors.
The other Frattini chief factors make up other layers of Frattini-ness. So much like the $p,p'$-series of a finite $p$-solvable group, you can view a finite group as made up of Frattini and non-Frattini layers of chief factors.
The Frattini factors can often be ignored in a group. For instance, if none of the Frattini chief factors are $p$-groups, then the group has no elements of order $p$. The centralizer of the $p$-chief factors is equal to the centralizer of the non-Frattini $p$-chief factors. Frattini factors are covered by all maximal subgroups, but each maximal subgroup in a finite solvable group picks out a chief factor that it complements. The Frattini factors are sort of a formless ooze between the more rigid layers of the complemented factors. There is a formula for the number of generators of a group and the number of generating sequences of a group in terms of the non-Frattini chief factors of the group.
The Frattini subgroup of the Frattini subgroup is the part of the Frattini subgroup that really doesn't matter, even to the part of the original group that doesn't matter. The Frattini subgroup is nilpotent, so its Frattini subgroup immediately devolves into the exponent-p derived series, which is pretty dull.
A: Perhaps we can see it the following way. We have the characterization of the Frattini subgroup of a group $\,G\,$:
$$x\in\Phi(G)\Longleftrightarrow\,\,\text{if for some}\,\,X\subset G\,\,,\,\langle\, X\,,\,x\,\rangle =G\Longrightarrow \langle\,X\,\rangle=G$$
as the non-generators of the group $\,G\,$.
Thus, if the series
$$\Phi(G)=:\Phi_0(G)\supset\Phi(\Phi(G))=:\Phi_1(G)\supset...\supset\Phi_n(G)=1$$
Now, what does it mean $\,\Phi(K)=1\,$? It means that any non-trivial element in $\,K\,$ is a non-redundant generator of some set of generators of $\,K\,$ . The only groups that I can think of right now fulfilling this are the elementary abelian ones, so perhaps we could say the Frattini Length of a group somehow measures how much of the original group we have to "scrap out" in order to be left with an abelian elementary one (or with any other kind of group for which any element is a non-redundant generator of some set of generators).
Just an idea...
