Self-similarity may originate from two origins: i.e., the process memory and the process' increments "infinite" variance. A distinction is attempted by employing the natural time chi . Concerning the first origin, we analyze recent data on seismic electric signals, which support the view that they exhibit infinitely ranged temporal correlations. Concerning the second, slowly driven systems that emit bursts of various energies E obeying the power-law distribution--i.e., P(E) approximately E(-gamma)--are studied. An interrelation between the exponent gamma and the variance kappa1(identical with <chi2> - <chi2>) is obtained for the shuffled (randomized) data. For real earthquake data, the most probable value of kappa1 of the shuffled data is found to be approximately equal to that of the original data, the difference most likely arising from temporal correlation. Finally, it is found that the differential entropy associated with the probability P(kappa1) maximizes for gamma around gamma approximately 1.6-1.7 , which is comparable to the value determined experimentally in diverse phenomena: e.g., solar flares, icequakes, dislocation glide in stressed single crystals of ice, etc. It also agrees with the b value in the Gutenberg-Richter law of earthquakes. In addition, the case of multiplicative cascades is studied in the natural time domain.