Hurst exponent with R

Hurst exponent, H, can be calculated from fractal dimension, D, as D = 2 - H.

I use the fractaldim package, available CRAN, to calculate fractal dimension. There is a complete description of the methodologies in Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data. University of Washington, Department of Statistics, Technical Report no. 577, http://www.stat.washington.edu/research/reports/2010/tr577.pdf.

With this package you can apply wavelet transforms, boxcount, etc. or the recommended madogram algorithm to calculate fractal dimension.

(Converted from a comment.)

This isn't exactly an answer, but

install.packages("sos")
library("sos")
findFn("hurst exponent")

should get you there pretty quickly. Notes: (1) you only need to do install.packages(...) once per installation of R, but library("sos") in every session; (2) you still need to work out whether the packages that you find this way are doing what you need -- but at least you know where to start.