Parameters `m` and `r`, involved in calculation of *approximate entropy (ApEn)* of time series, are *window (sequence) length* and *tolerance (filter value)*, correspondingly. In fact, in terms of `m`, `r` as well as `N` (number of data points), *ApEn* is **defined** as "natural logarithm of the relative prevalence of repetitive patterns of length `m` as compared with those of length `m + 1`" (Balasis, Daglis, Anastasiadis & Eftaxias, 2011, p. 215):

$$ ApEn(m, r, N) = \Phi^m(r) - \Phi^{m+1}(r), $$

$\text{where }$

$$ \Phi^m(r) = {\LARGE{\Sigma}_i} lnC^m_i(r)/(N - m + 1) $$

Therefore, it appears that changing the tolerance `r` allows to control the (temporal) *granularity* of determining time series' entropy. Nevertheless, using the default values for both `m` and `r` parameters in `pracma` package's entropy function calls works fine. The only **fix** that needs to be done to see the *correct entropy values relation* for all three time series (lower entropy for more well-defined series, higher entropy for more random data) is to *increase the length of random data vector*:

     library(pracma)
     set.seed(10)
     all.series <- list(series1 = AirPassengers,
                        series2 = sunspot.year,
                        series3 = rnorm(500)) # <== size increased
     sapply(all.series, approx_entropy)
      series1   series2   series3 
      0.5157758 0.7622430 1.4741971 

The results are as expected - as the predictability of fluctuations decreases from most determined `series1` to most random `series 3`, their entropy consequently increases: `ApEn(series1) < ApEn(series2) < ApEn(series3)`.

In regard to other **measures** of *forecastability*, you may want to check *mean absolute scaled errors (MASE)* - see [this discussion](http://stats.stackexchange.com/q/23007/31372) for more details. *Forecastable component analysis* also seems to be an interesting and new **approach** to determining forecastability of time series. And, expectedly, there is an `R` package for that, as well - [**ForeCA**](http://cran.r-project.org/web/packages/ForeCA).

  
    library(ForeCA)
    sapply(all.series,
           Omega, spectrum.control = list(method = "wosa"))
     series1   series2   series3 
     41.239218 25.333105  1.171738 

Here $\Omega \in [0, 1]$ is a measure of forecastability where $\Omega(white noise) = 0\%$ and $\Omega(sinusoid) = 100 \%$.

**References**

Balasis, G., Daglis, I. A., Anastasiadis, A., & Eftaxias, K. (2011). Detection of dynamical complexity changes in Dst time sSeries using entropy concepts and rescaled range analysis. In W. Liu and M. Fujimoto (Eds.), *The Dynamic Magnetosphere, IAGA Special Sopron Book, Series 3, 211.* doi:10.1007/978-94-007-0501-2_12. Springer. Retrieved from http://members.noa.gr/anastasi/papers/B29.pdf

Georg M. Goerg (2013): Forecastable Component Analysis. JMLR, W&CP (2) 2013: 64-72. http://machinelearning.wustl.edu/mlpapers/papers/goerg13