A Repulsor Beam
Perhaps not the fastest, but a VERY fast form of propulsion was that used by the Aldeans. They managed to transport the Enterprise-D 3 days away at warp 9 almost instantaneously:
RADUE [on viewscreen]: Captain, we want you to understand the nature of your choice. A small demonstration of our power.
(Something whizzes around the planet and knocks the Enterprise spinning out of orbit, into deep space)
PICARD: What was that?
DATA: I believe it was a repulsor beam.
PICARD: Position report.
LAFORGE: This is unbelievable, sir. According to my calculations, we're three days from Aldea. At warp nine.
RIKER: And they call that a small demonstration?
PICARD: Geordi, get us back to Aldea. Warp nine.
LAFORGE: Aye, sir.
RADUE [on viewscreen]: Captain, if you don't accept our terms, the Enterprise will be pushed so far away that by the time you return, your children will be grandparents.
(TNG 'When the Bough Breaks')
After doing some calculations, this is a speed of approximately 3.93 x10^12km/s (assuming it took them about 3 seconds to travel that distance) i.e. 3,930,000,000,000 km per second
Whether or not this was a 'small demonstration' or not is up for debate, but considering their technological feats of being able to cloak an entire planet, it wouldn't at all surprise me if they could push them that far away.
This is pretty fast indeed!