We present a case study of the large-scale “fractal” behavior of concrete families of random processes that arise in complex systems. Among other things we will exhibit two random functions both of which are “multifractal” on large scales, but only one of which shows “intermittency.” This contradicts  the commonly-held view that “multifractality” and “intermittency” can be used interchangeably. This is based on joint work with K. Kim and Y. Xiao.