The fractal nature of a nanofibrous network has been discovered by in-situ measurements on fibers of N-lauroyl-L-glutamic acid di-n-butylamide, which grow in isostearyl alcohol upon cooling; critical parameters are derived and discussed. Branching induced by crystallographic mismatch was identified as the nanofibers' growth mechanism (see Figure for an SEM image; bifurcation positions are highlighted by dotted circles).