I have a vague recollection of a ship in the Animated Series travelling at warp 36 or so. Some people don’t like using the Animated Series as an example, though.
Warp in general wasn’t always treated with a certain… consistency. In the old Trek Technical Manual, warp was cubed to describe the multiple of the speed of light. So, warp 2 was 2x2x2 = 8 times the speed of light. Warp 7 was 7x7x7 = 343 times the speed of light and so forth.
In Next Generation, that was changed to some sort of asymptote. So, warp speed was some multiple of the speed of light, but the maximum value in the universe could be warp 10 (but it was an asymptote, so it took infinite energy to reach that level – unless Wesley was diddling with the widget banks), which would be a lot like using an Infinite Improbability Drive, except no whales.
Unless you were travelling back in time specifically *for *whales, in which case, no bowl of petunias.
By the ship’s transporters, usually about orbital level. Other creatures, species, and miscellaneous hyperintelligent glowing post-production effects have been able to teleport people much much farther. Kirk and Co. were transported many light years to Triskelion to fight, and Gary Seven was mid-beam to Earth from a hugely far away place.
The biggest I can recall is the mass of lots of people and their gear. The transporters we saw in the series had six pads, so if you figure 100 kilos per person, that’s about 600 kilos. Round up to say 800 kilos to account for excessive makeup, maybe?
In the Technical Manual and the blueprints, there are cargo transporters, which don’t have individual pads, but big transporter “areas”. I can’t recall those having a mass-limit specified, though.
It would be easier to ask which can’t. Star Trek is a very promiscuous universe and most anyone can comingle with anyone else. Offhand, I can’t recall seeing an Andorian/whatever hybrid, or a Horta/whatever hybrid, but I blame Bad Attitude for the former and, well, the latter, too. Plus, you never know what might happen to a Horta in the middle of orgasm, but I can’t imagine anyone wanting their private parts near it.
Yep, in one episode of Next Generation, Scotty has discovered a Dyson Sphere and locked himself there in a transporter matrix. Oddly enough, however logical Dyson Spheres are (and relatively simple to construct, compared to some of the constructs I’ve seen in the show), I can’t recall mention of them in any of the series.
Never saw a ringworld, either, or a discworld, but I suspect those are, er, copyrighted.