A Physics Quiz of a different type

[DonRotolo]

Someone else offered an explanation as well:

As you gain energy, the mass of the object increases, because energy has weight. This includes kinetic energy. So as speed increases, so does mass, requiring more and more energy to maintain acceleration. At some point, it gets really massy, such that you can't push it hard enough to accelerate it.

I am told that this "weight gain with energy" has actually been measured, but I can't find that reference.

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OK, does a photon have mass? If yes, then it cannot travel at the speed of light, but since it does, at rest it must be massless. (Yes, it has momentum, but that's a special case. In theory you can approach the speed of light using a flashlight as your propulsion unit. Might take a while though...)

Are there any nonzero-mass particles/objects that travel at c? Can there be?