If photons lose frequency (due to redshift) can they reach 0 Hz? What would happen then? I haven't got much knowledge about this topic and I would like to know it.
 A: Yes, they can, or rather the photon source can vanish behind be hidden behind an event horizon (cf black holes in case of gravitational redshift, the cosmic event horizon in case of cosmological redshift due to the metric expansion of space or the Rindler horizon in case of acceleration; for any finite relative velocity between source and observer, the redshift will of course stay finite, though...).
To clarify, 0 Hz photons do not exist (0 Hz means no energy, ie the electromagnetic field remains undisturbed) and thus obviously cannot be measured, but our theories predict a continuous increase in redshift, and the event horizon is the place where it goes to infinity.
A: There is little mystery. 
In special relativity, as @peterh said, it goes to infinity only for an observer going at speed c with respect to the source. Only massless particles can do that, so you can't be such an observer.
In general relativity you also can't. Two cases that shows it, one near a black hole, and one the event horizon. 
The cosmic event horizon is discussed in the answer by Christoph. At the cosmic event horizon the light emitted now cannot reach us. The universe is expanding faster than c, and anything that happens NOW (in comoving time) beyond that we can't see, it can not affect us. From Wikipedia,  that is about 5 Gpc. Our particle horizon is different, and so is the Hubble horizon, see the wiki article 
https://en.wikipedia.org/wiki/List_of_cosmological_horizons#Event_horizon
Another horizon, an event horizon, can be obtained for the Schwarzschild spherically symmetric (and you can for Kerr also), and it is, and you can see it at https://en.wikipedia.org/wiki/Redshift. The redshit for light emitted there would be infinite. In fact the event horizon is where any light emitted near there, inside the black hole, going out, ends up. The horizon is a light like surface. At the Schwarzschild black hole horizon the redshift is infinite, and light cannot escape.  
