From: Interpreting Residual and Null Deviance in GLM R we see that the residual and null deviances are calculated from the baseline log-likelihood value given by the saturated model.
It also notes that the smaller these deviances are, the better the fit the model is to the data. However, this effect can also be attained by just comparing the fitted and null model to a theoretical (perfect) model having a perfect fit to the data. Note that in this case likelihood-ratio tests are also unaffected.
So in what situations is the saturated model giving more information than the theoretical model with perfect fit? The only reason I can think of is given by the comment to this answer, but in this case isn't it more sensible to compare the log-likelihoods of the saturated model to a perfect model?