(Following answer does not treat TAS as canon)
All Good Things had an alternate future Enterprise D travel at warp 13, which should be faster than warp 15 from TOS (5165.88 *c and 3,375 *c, respectively), given the scale recalibration, if we assumed a linear function for warp (which it isn't, so warp 13 is actually much much much faster than that). However (as reported in the Memory Alpha article), warp 13 in that future would have been regarded as warp 9.95 in regular-time TNG scale...which is, unfortunately, slower than the TOS 15, as warp 9.975 is only 3053 *c (according to VOY). VOY's tech manual states that 9.99 is 7912 times the speed of light, which would be 19.92~ in TOS speeds.
So! What this does mean is, according to the speeds stated in the VOY tech manuals, the Enterprise D has reached the highest warp speed yet when it was thrown through space by the traveler, traveling at Warp 9.9999999996 (according to the TNG tech manual). And that's it? The fastest warp is less than 10, using a different scale and exponential function that I can't find.
However! Nothing (afaik) on the show in TNG or VOY was stated at traveling above 9.98~ish (which should be faster than TOS 15), so @rand al'thor has the best answer as the lightning bolt from TOS.
TOS: v = (warpFactor to the power of 3) * c
TNG: v = if warpFactor <= 9 then (warpFactor to the power of 10/3) * c