  We consider exponential large deviations estimates for unbounded observables
on uniformly expanding dynamical systems. We show that uniform expansion does
not imply the existence of a rate function for unbounded observables no matter
the tail behavior of the cumulative distribution function. We give examples of
unbounded observables with exponential decay of autocorrelations, exponential
decay under the transfer operator in each $L^p$, $1\le p < \infty$, and
strictly stretched exponential large deviation. For observables of form $|\log
d(x,p)|^{\alpha}$, $p$ periodic, on uniformly expanding systems we give the
precise stretched exponential decay rate. We also show that a classical example
in the literature of a bounded observable with exponential decay of
autocorrelations yet with no rate function is degenerate as the observable is a
coboundary.
