Doppler effect, mass changes of objects, and all relativistic effects (v^2/c^2) are based on $c$ as the maximum velocity ($Vt$) in the Universe. We have a fairly good understanding of relativity (I will stop at SR). If $Vt$ was only slightly higher than $c$ (say 0.001%), then certain conflicts in physics could be resolved. For example, photons could then have a very insignificant rest mass, which would then match all the other photon physical characteristics and properties.
Therefore my question is - What is the experimental and/ or measured data for setting that ultimate velocity limit? Keep in mind that assuming 'nothing goes faster than the speed of light' is not what I am asking as it is generally agreed that we have not seen anything yet. Nor am I asking for mathematical assumptions that were then incorporated into the interpretation of the measured data, as this is circular logic.