What is the fastest speed that a massive object can travel at? What is the fastest speed (in miles per hour) that a massive object can travel at? I have heard that an object can travel at the speed of light, but I've also heard that massive objects cannot travel at the speed of light. 
 A: Well, by massive, I assume you mean objects that have non-zero rest mass. In that case, it would take infinite energy for that object to reach the speed of light. However, their speed would get closer and closer to the speed of light as more energy is put in, until their speed was practically (but not exactly) the speed of light. Additionally, the smaller the rest mass, the easier it is to accelerate them closer to the speed of light. Neutrinos, which have the smallest non-zero rest mass that we know of, travel at speed so close to the speed of light that we have not actually been able directly measure a neutrino traveling slower than the speed of light (that is, the speed of light has been within the margin of error of the measurement), and for a while, they were thought to be massless. Why we know they aren't massless is outside of the scope of this answer. The fastest proton that we've observed is the Oh-My-God Particle, which was going at $100-9.88*10^{-18}$% of the speed of light, or only $3*10^{-15}$ miles per hour less than the speed of light, and had the energy of a baseball pitched by a professional pitcher. This makes it the particle with the greatest total energy to rest mass the we've observed.
A: In theory, a massive particle can be accelerated asymptotically close to the speed of light.
But one can never actually reach the speed of light because the kinetic energy of such a (massive) particle would be infinite.
This is because the mass of the particle itself become heavier at higher speeds due to relativistic effects. Specifically:
$$\text{Kinetic Energy} = (\gamma -1) \times mv^2$$
Where $$\gamma = \frac{1}{\sqrt{1-(v^2/c^2)}}$$
You can see that when $v = c$, $\gamma$ goes to infinity.
