As an example, consider a variety of generalized linear models -- for example, gamma, Poisson or inverse Gaussian regression models. Those models (plus some others) correspond to models for data that could be classified as ratio data (the zero is meaningful, 2 really is twice as much as 1, etc).
Further, these models are not equivariant to adding or subtracting constants to/from the data (so it's plainly not simply interval).
If you're trying to analyze some kinds of times/counts/monetary data -- and numerous other kinds of data, such models may be very useful.
In a related fashion, a log-transform or a power transform isn't generally meaningful for data with an arbitrary zero (where the data mean exactly the same thing with a different zero) even if the data happen to be all-positive. It is often meaningful for ratio data.
One should also be careful not to get too hung up on Stephens' typology in particular; there are other typologies that can be more useful for statistical purposes; indeed the notion of using typologies to decide analysis (rather than the question the data are being used to investigate or answer) may often be misplaced. It can sometimes be helpful, but not infrequently, it's the opposite.