Do massless particles really exist?

Here is a quick & simple answer until the professionals arrive. In the Standard Model, it is zero. This $< 1\cdot 10^{-18} \frac{\mathrm{eV}}{c^2}$ is an experimental upper limit (i.e. if it has a rest mass, because of physics beyond the Standard Model, it must be smaller than this value).

This value is very small, compared to the estimated rest mass of the neutrinos (which is of the order of some tenths of an $\mathrm{eV}$).

We can't measure to infinite precision; so even if a particle had in fact zero mass we couldn't experimentally measure it to the infinite precision needed to justify this; which is why certain amount of judgement is called for, and that judgement is made in the context of a theoretical framework.

The second point to make is that all particles with zero rest mass travel at the speed of light and they have momentum due to this motion.

As one answer has pointed out already such particles are gauge bosons which mediate the weak, strong & EM forces. For the EM force, this is the photon.

There are indeed massless particles.

As of 2015 there were two known massless particles (both gauge bosons): the photon (carrier of electromagnetism) and the gluon (carrier of the strong force). It should be noted, however, that gluons are never observed as free particles, since they are confined within hadrons.

Gravitons (if discovered) would be another massless particle.

Of course, it must be kept in mind that nothing can be measured to infinite precision. Because of this, we will never measure a photon's rest mass and find it to be zero. As our measurements get better and better, it will get closer to zero, but it will never quite get there.

Interestingly, according to this website, scientists are able to look at Coulomb's Law and other experimental results and place upper bounds on what the photon's rest mass can be measured as. The best upper bound to date is $1.07×10^{−27}$ atomic mass units. The equivalent of this is what you saw on the Wikipedia sidebar.

There's more information at this website about massless particles.