We are stuck on the idea that 300,000 kilometres a second is a speed limit, because we intuitively believe that time runs at a constant universal rate. However, we have proven in many different experimental tests that time clearly does not run at a constant rate between different frames of reference. So with the right technology, you can sit in your star-drive spacecraft and make a quick cup of tea while eons pass by outside. It’s not about speed, it’s about reducing your personal travel time between two distant points. And that has a natural limit – of zero.
A decent little tutorial. One approach that can also be used is to note that the invariant speed, given by the velocity four-vector, has a constant length of c. The three spatial components are the velocity and the time component is \(gamma\)ct. If you are stationary, as we all are in our own rest frame, time ticks normally. But when one is moving — the spatial velocity vector is nonzero — the time component of the vector shortens compensates.
Update: Chad points out some sloppiness on my part. The vector is given by \({v_x}^2+{v_y}^2+{v_z}^2-gamma c^2\) so the components get bigger while the resultant stays constant.