I know Goku has the whole instant transmission thing (would would appear to obviously break FTL, perhaps to an arbitrary degree -- I don't remember if that thing had range limits or anything.
Yes, he's used it to travel not only between planets orbiting different stars, but also to instantly travel between either galaxies or dimensions depending on which translation you consider. There doesn't appear to be any range limit. But it's clearly a
teleport, not a "speed" thing.
but I don't remember if he could break the limit just via pure speed
any examples off the top of your head?
The difficulty in evaluating Goku has always been that a number of his feats are depandant on inferring what he's capable of to what other characters are actually demonstrated to do, and in general require a lot of speculation and mental gymnastics.
For example, Vegita is shown in the anime both to be able to casually destroy planets as well as to travel between solar systems under his own power. And Goku is shown to be both faster and able to put out more power than Vegita on many occasions. Therefore we
infer that he is capable of these things as well even though he's never shown doing them personally. Same things goes with breathing in space. There are lots of examples of people of his species flying around in space without difficulty, but Goku himself is never ever once shown to do it himself. So do we conclude that he can't just because he's never shown to, or do we conclude that because member of his species with far less ability are able to, that he can as well?
Unneceesarily complicated example:
http://view.thespectrum.net/series/dragon-ball-volume-01.html?ch=Volume+40&page=124That's 5 loops around the earth. Measureing with a straightedge, those paths are about 1.8 times the diameter of the planet. So, 5 loops 1.8 times a 12,756km diamater works out to:
5 * 2 * pi * (1.8 * 12756 / 2)^2 = 4,138,218,560km
That was done in "under a minute." We don't know how much under a minute, so let's just assume 60 seconds for simplicity. So divide by 60 and we get 68,970,309 km/minute The Speed of light is 17,987,547 km/minute. Gotenks was travelling on the order of 3.8 times the speed of light. Goku is faster than Gotenks, therefore Goku can travel faster than the speed of light. There's no indicator of how
long he can maintain that speed.
But we have to make lot of assumptions. For example, Goku is able to catch Gotenks, but is Gotenks going as fast as he was shown above at the time? Maybe he was tired and therefore going more slowly. We don't really know.
A simpler example would be the training he did with Mr. Popo. He was explicity shown to be able to travel faster than lightning. which is on the order of one third to one half the speed of light. Kaioken is explicity stated to increase both strength and speed by its multiple. And he's done kaioken as high as 20x. So 20 * one third the speed of light would be about 6c.
There are other example, but so far a I know they all require similar amounts of math and or speculation. The screwball attack estimate based on the Serpent's path feat put him at 2.3c. And I think that was quite some time prior to the first time he managed kaioken as high as 20, so that seems reasonable. But on the other hand his training with Mr Popo was when he was just a kid. So if we've established a lower bound of at least 6c when he was a kid, obviously he's able to move faster as an adult, but how much faster? If you apply some of the extreme multiplliers that some people are coming up with for his various forms you could possibly claim a speed on the order of many thousands of c. Though that would be an extreme outlier compared to the majority of examples which put him at about 1-10 c.
So yes...definitely faster than light, but how much faster is debatable. It's also debatable how
long he can maintain those speeds. For example, if I recall correctly the "faster than lightning" bursts were to travel distances of at most maybe a couple dozen feet. Travelling 50 feet at 6c is very different from maintaining 6c for hours or weeks at a time to travel between planets. But then on the other hand, using the screwball attack example, he maintained their estimated 2.3c for 28 hours.