Dimensionless numbers or parameters that are $\ll 1$ or $\gg 1$ Many times, in various fields, authors use the notation:
$$C\ll 1$$
or
$$C\gg 1,$$
where $C$ is some parameter related to the system being studied. I know this is highly dependent on application, but is there a general range of $C$ that satisfies the relations above?
I'm asking because I have a parameter that characterizes two types of behaviors of my system. For $C = 0.074, 0.005, 0.086$ type A behavior is observed and for $C = 0.78, 1.24$ type B behavior is observed.
If I say type A behavior is observed for $C \ll 1$, is that consistent with the data presented above?
 A: This is a complement to dmckee's answer by way of an example.
Suppose that I want to determine the acceleration due to gravity of an object near the surface of the earth.  Let $h$ be the height of the object above the surface, then according to Newton's Law of Gravitation, I get
\begin{align}
  a = \frac{GM}{(R+h)^2} = \frac{GM}{R^2}\left(1-2\frac{h}{R}+O\left(\frac{h}{R}\right)^2\right)
\end{align}
Where $R$ is the Earth's radius, $G$ is Newton's gravitational constant, and $M$ is the mass of the Earth.  Recognizing the coefficient in front of the Taylor expansion as (essentially) what we usually call $g$, and calling the dimensionless ratio $h/R$ of the height above the earth to the Earth's radius $\epsilon$, we find that
\begin{align}
  a = g(1-2\epsilon+O(\epsilon^2))
\end{align}
Notice that this expression has the following properties:


*

*When $\epsilon = 0$ the acceleration is $g$.

*When $\epsilon$ is close to $0$, the acceleration is close to $g$.
In the above situation, I would be comfortable saying that

If $\epsilon \ll 1$, then the acceleration due to gravity is $g$.

If I am making a measurement, then we can also make the notion of $\epsilon\ll 1$ sharp depending on the limitations of our instrumentation.  Suppose that I am measuring the acceleration due to gravity above the surface of the Earth with an instrument that can measure deviations in $g$ up to $10^{-3}\,\mathrm{m}/\mathrm s^2$, then here I might be inclined to say that

My measurement device will adequately measure deviations in $g$ provided $\epsilon \ll 1$.

but what I really mean here is

My measurement device will adequately measure deviations in $g$ provided $\epsilon < 10^{-3}/2$.

a less ambiguous statement.  The main point is that use of the $\ll$ symbol is context-dependent, as emphasized in dmckee's answer.
A: Generally taking note of these relationships is a precursor to either (a) applying an approximation or (b) using a purturbative or series solution. 
In case (a) what qualifies is completely a matter of your sensitivity to error. If you are going to throw out terms $\mathcal{O}(C^{\pm 2})$ and require a 1% approximation then $C$ had better differ from 1 by a factor slightly larger than 10.
In case (b) there may be hard limits on the value of $C$ for which the mathematical approach you are using converges. You must understand the analysis that follow or you are sunk. Once convergence is guaranteed we come back to the situation in case (a): you look at the residual after you are done summing as many orders as you are going to work and compare that to your desired sensitivity.

As a rule of thumb, factors less than 10 rarely qualify for "much [greater|less] than".
