The Bandt-Pompe permutation entropy and the Jensen-Shannon statistical complexity are used to analyze fluctuating time series of three different turbulent plasmas: the magnetohydrodynamic (MHD) turbulence in the plasma wind tunnel of the Swarthmore Spheromak Experiment (SSX), drift-wave turbulence of ion saturation current fluctuations in the edge of the Large Plasma Device (LAPD) and fully-developed turbulent magnetic fluctuations of the solar wind taken from the Wind spacecraft. The entropy and complexity values are presented as coordinates on the CH plane for comparison among the different plasma environments and other fluctuation models. The solar wind is found to have the highest permutation entropy and lowest statistical complexity of the three data sets analyzed. Both laboratory data sets have larger values of statistical complexity, suggesting these systems have fewer degrees of freedom in their fluctuations, with SSX magnetic fluctuations having slightly less complexity than the LAPD edge Isat. The CH plane coordinates are compared to the shape and distribution of a spectral decomposition of the waveforms. These results suggest that fully developed turbulence (solar wind) occupies the lower-right region of the CH plane, and that other plasma systems considered to be turbulent have less permutation entropy and more statistical complexity. This paper presents the first use of this statistical analysis tool on solar wind plasma, as well as on an MHD turbulent experimental plasma.