How much does the full-size LEGO Bugatti cost? I read that the LEGO developers have built a real driving Bugatti from Lego Bricks.

Did they only use original LEGO parts? Which ones and how many? Which sets does one need to buy and how many? And how much would that cost?

And bonus question, is there a "legal" (LEGO only) way to make it faster?
From Top gear's article on it:


  This was laid over a steel frame; one of three ‘parts’ that aren’t actually Lego (the others being the real Chiron wheels and a steel roll cage).


So no, not only original LEGO parts. I don't think (and can't see why they should have) kept track of which parts they used, so noone can really answer the rest of your questions, but from the figures we know (for instance from the linked article):
- 1 million Technic pieces
- 1,5 tonne
- 2304 motors

we can estimate the price. As a significant part of the pieces are probably in the expensive end (an average weight of 1,5 gram matches that), 1/3€ is probably not completely off as an average piece price. Motors are significantly more expensive, they haven't said (or I haven't read it at least) what kind of LEGO motors they use, but they cost 10-20€ a piece, giving a total for the motors of at least 20.000€, so in total the car might only (compared to a real Bugatti Chiron) cost ~350.000€, and then there's the "13,500 man hours", salaries vary a lot, and if you were to replicate it you would actually have to pay it, just spend the time, and it is more than 1,5 year (working 24 hours/day).

Edit: I forgot the bonus question. The 2304 motors they used take up a bit of space, and typically LEGO motors aren't manufactured to have a very precise speed, so when connecting more I would think a differential would be need for each (except the first), they also take up space. If you could fit more motors (and differentials) in there, you would theoretically increase its power, as things are I think (but the physics is not really my area) it's hard to tell if it would manifest itself as more speed and if such how much, but it wouldn't be impressive. I don't see any other ways to increase the speed than adding more motors.