Q:

Is there absolute proof that an object cannot exceed the speed of light?

Have any known experiments ruled out travelling faster than the speed of light?
Or is this just a widely accepted theory?

A:

This needs to be emphasized : 
No theory, and Special Relativity that is used in physics to describe data is a theory,  can be proven, in the way a mathematical theory can be proven with a QED at the end.Theories can only be validated by the agreement with the data on the predictions they make.
That is why the OPERA faster than light neutrinos  (wrong) observation gripped our attention.
Even one falsification of a theory's predictions invalidates a theory and a new axiomatic formulation, a new theory, would have to be found .
There has been no falsification by experiment of special relativity.
That is the difference between physics and mathematics.

A:

I think the comments are explaining it just right...

Is there absolute proof that an object cannot exceed the speed of light?

No, there isn't. Similar to the statement that "there's no such thing as 100% efficiency", there doesn't exist an absolute proof to anything (especially in Physics)

Have any known experiments ruled out traveling faster than the speed of light?

No again. The speed of light and the particles traveling closer to speed of light are experimentally verified by the consequences like time dilation (for e.g. Muons). But, there are a lot of theories regarding the FTL because physicists showed a lot of interest back at 1950s on the subject "Tachyons"...

Or is this just a widely accepted theory?

SR is accepted widely because of the verifications it provided and the predictions it made. And for this reason, SR is most widely accepted all around. In fact, it's more useful in particle physics when accelerating particles to high energies.

