  We show that detection of single photons is not subject to the fundamental
limitations that accompany quantum linear amplification of bosonic mode
amplitudes, even though a photodetector does amplify a few-photon input signal
to a macroscopic output signal. Alternative limits are derived for
\emph{nonlinear} photon-number amplification schemes with optimistic
implications for single-photon detection. Four commutator-preserving
transformations are presented: one idealized (which is optimal) and three more
realistic (less than optimal). Our description makes clear that nonlinear
amplification takes place, in general, at a different frequency $\omega'$ than
the frequency $\omega$ of the input photons. This can be exploited to suppress
thermal noise even further up to a fundamental limit imposed by amplification
into a single bosonic mode. A practical example that fits our description very
well is electron-shelving.
