There's only one thing that is considered to be massless and that is the photon (well actually also theorized particles such as gluons and gravitons would be massless as well, but these particles are not observed).
The fact that photons are considered to be massless is not something that we observe, rather it is a consequence of the theory of relativity (along with photoelectric effect which says that light CAN act as a particle as well as a wave).
I don't think this answers your question, but I don't really know how else to answer it (because what you are saying isn't really what the current theory says).
Edit:
So although I don't think your question is well thought out, that DOESN'T mean that you aren't asking a good question.
The biggest problem in your question, is how exactly do we even measure (or observe) that something has mass?
Seems obvious when you look at everyday life (because EVERYTHING has mass in our everyday lives). But once you go to sub-atomic particles it become much more difficult (can't exactly put an electron on a scale).
So how would we know whether or not something has mass?
First guess might simply be, if it interacts with gravity, then it must have mass. Problem with that is that photons interact with gravity (Doppler effect), yet we still say they have no mass!
This is explained through General Relativity--but this is my point. An observer might say, well the photon gets "accelerated" through a gravitational field (because the wavelength gets shorter--i.e. higher energy), thus the photon must have some mass. However, our theory of gravity (General Relativity) can explain why this happens, yet the photon is still massless.
So, a full discussion of the full implications of your question would take several pages and several papers. In some sense it boils down to "reality" versus "perception". Because the idea of mass is so ingrained into our conscience, it's difficult to think of mass as something imaginary (i.e. made up)--but, in fact, the whole idea of mass, charges, energy, wavelengths, etc. are mathematical constructs that we have made up to explain nature.
Example: when you lift a heavy object, surely it has more mass, right? How do you know that? You aren't observing it's mass, you are observing the fact that the force required to lift it is higher--you are saying that it has more mass because the theory of gravity says that the more massive an object is, the more it is affected by gravity. Since it requires more force to lift, it must be more massive.
I hesitate to say this because it's easy to get misconstrued by non-technical people, but most people (and even a lot of physicists) don't understand how artificial the theories of physics are. Just because you have a theory which explains nature extremely well does NOT mean that you are describing what is ACTUALLY going on in nature.
A perfect example of this is the idea that Greeks thought the Earth was the center of the universe. This was the theory and it gave us the name "planet" (wandering star). The planets had strange trajectories under this theory--yet through a complicated system of epicycles, the ancients WERE able to predict the trajectories of the planets. The theory made good predictions about nature, yet it's obvious now that this theory was NOT describing what was actually happening.