Why would we want to calculate the Lyapunov exponent for experimental data? Searching Google Scholar for "Lyapunov exponent from time series" turns up multiple papers (some of them highly cited) suggesting methods for estimating the largest Lyapunov exponent or sometimes even the whole Lyapunov spectrum from experimental time series.  
None of these methods seem to be very precise, (as can be verified on synthetic data, which I have), and even if viewed as a binary classifier (i.e. used to determine if a system is chaotic/regular based on if the maximal Lyapunov exponent is positive/negative) yield both false negative and false positive results. 
Do any of these methods have any practical (rather than theoretical) use?
 A: In control theory, Lyapunov exponents correspond to the eigenvalues of the linearized nonlinear dynamical system which provide local information about the stability margin. This information can be used to design stabilizing controllers as in this paper.
A: As you already noted, a positive Lyapunov exponent (if properly ensured with surrogates, etc.) can tell you whether a system is chaotic. However, the differentiation to make is not only between chaos and regular dynamics¹, but between chaos and a stochastic dynamics². Somewhat simplified you want to distinguish between a finite positive and an infinite Lyapunov exponent.
Now, what is the practical value of this?


*

*If we know the system is chaotic and not stochastic, we may manipulate it to some extent, e.g., using chaos control.

*A chaotic system can be predicted to some extent, which in turn may allow for targeted interventions.
The Lyapunov exponent tells us how far we can predict the system with a given knowledge about it.
For example, the Lyapunov exponent of the weather gives us a natural limit of weather forecasts.

*It guides us on how to approach modelling a system, more precisely on whether to use a stochastic or deterministic model.
This is admittedly not very practical, but the models in turn may help us understand a system and have a lot of practical implications.

*If we know that we are experimenting with a chaotic system, we can have some expectations on how replicates of the same setup will behave.


¹ If you really have a regular dynamics, it’s usually pretty obvious.
² More precisely:
A dynamics dominated by inaccessible stochastic components.
Most real systems have stochastic and deterministic components, where stochastic merely means that it is beyond the scope of our models and measurements, e.g., molecular billiard when investigating a chemical reaction in a tube.
