First of all, just like this video title suggests, the all famous E=MC^{2} formula is just incomplete. The complete formula goes something like this:

E^{2} = (MC^{2})^{2} + (PC)^{2}

C in both terms is the speed of light, M is the object's mass and P the object's momentum. One that know geometry will immediately "see" the 3 terms of the right triangle, as described by the Pythagorean theorem. According to this theorem which applies to all right triangles, the square of the hypotenuses equals to the sum of squares of the perpendicular sides. To our Einstein's equation, E^{2} represents the square of the hypotenuses and the terms (MC^{2})^{2} and (PC)^{2} represent the two perpendicular sizes.

Having this in mind, let's now examine what happens to a speeding object. If V is the object's velocity, the following equation arises:

V = C x (PC/E)

It is obvious that if PC was equal to E, then the term PC/E would become E so the above equation would become

V = C x 1 => V = C

Which means that the speed of the object would be equal to the speed of light.

BUT

From the right triangle previously studied it is clear that E can never become equal to PC for an object that has mass. It could ONLY be true if M was 0:

This final step was not really necessary because from geometry we know that:
"The length of hypotenuses is always longer than the individual lengths of the two perpendicular sides"

Which means that for an object with mass, regardless of how small this mass is, E will always be bigger than PC, thus PC/E will always be smaller than 1, thus V will always be smaller than C

@Panagiotis Kalogeris Yep.... we can reach up to 99.999999999% but never 100%...
.. UNLESS ..
of course if these people in CERN do something weird with this Higgs field and mass is disappeared. If mass=0, MC^2 becomes 0, so PC equals to E, so with little energy we run like crazy...