What is the fastest process or shortest time in nature? We know about some events that happen very quickly.  For example, the dielectric relaxation time is about $10^{-14}\, \mathrm{seconds}$. 
I'm interested in other processes that switch extremely fast or or could be use as a very short tick in a clock.
 A: The fastest fully localized process is the evaporation of the smallest black hole worth the time - it takes one Planck time or so, $10^{-43}$ seconds. There are other characteristic processes in quantum gravity that take a Planck time - the shortest time scale for which the usual spacetime geometry works.
However, if you allow changes in collective properties of large objects, there are much shorter times. For example, the mass of the visible Universe is $3\times 10^{52}$ kilograms or so. The corresponding energy, via $E=mc^2$, is $10^{61}$ Joules. The periodicity of the corresponding quantum wave, via $E=\hbar\omega$, is about $10^{-95}$ seconds. So the wave function of the Universe periodically changes its phase $10^{95}$ times every second.
I guess that you mean local processes, and moreover some processes accessible experimentally. That's a question with no permanent answer. The whole field of particle physics may be classified according to the time scale we can resolve. The time scale you mention is that of atomic physics; the time scales in nuclear physics are up to 10 orders of magnitude shorter, $10^{-24}$ seconds or so. The LHC is probing time scales that are 4 orders of magnitude shorter than that. In principle, this progress could continue. The "higher energy" we have, the shorter times (and distances) we may resolve.
A: a photon traveling a planck length.
