Tsallis's is the most simple generalization that you can imagine. It is mathematically handy. In nonextensive situations, correlations between individual constituents in the system do not die off exponentially with distance as they do in extensive cases. Instead, the correlations die off as the distance is raised to some empirically derived or theoretically deduced power, which is called a power law. If a plot of the logarithm of the number of times a certain property value is found against the logarithm of the value itself results in a straight line, the relationship is a power law.
Tsallis entropy is applicable to hundreds of nonextensive systems with such power-law scaling.
Power laws are helpful in describing not only fractal behavior but many other physical phenomena as well. Unfortunately, Tsallis has no proof from first principles that his expression of entropic nonextensivity is the best one possible. Michel Baranger, an emeritus professor of physics at MIT, agrees that the lack of such a proof has led many physicists to be skeptical. But I would still like to see justification of this formula from first principles. What he did, intuitively, was really remarkable.