The Complete Guide to Anachronisms in Samurai Champloo--
Bogus Booty
Episode 15: Bogus Booty
Again a stylistic device rather than an anachronism proper, but deserves a mention: ChamplooMasta and others mentioned "the juxtaposition of the old-skool (Shakespearean style) Japanese singing narration at the beginning of Ep 15 with the turntable scratches".

Vroom!--Neko-san (first) and Laura Bryannan (second) both noted that Mugen in pursuit of Yatsuha spreads his arms and makes "zoom" noises like a kid imitating an airplane. Not only is this totally anachronistic, it's totally fkn' adorable. =)
Shu mentioned (and dakameleon provided this image) that Tsutenkaku Tower, an Osaka landmark, appears in this episode: it wasn't built till 1912.

The power source of the brothel. A whole big can of worms here. First of all, there's that elevator. While primitive animal- or man-powered lifts and hoists had been in use for ages--there was even one in the Roman Coliseum-- this looks and operates like an Otis-style hydraulic or at least counterweighted elevator, of which there's no known example earlier than 1743 (Louis XV commissioned it for his personal chambers in the Palace of Versailles). The hydraulic elevator is generally said to have been invented in the 1800s and was firmly established by Elisha Otis' invention in 1853 of the safety brake. Using the Versailles example, the brothel's lift is just marginally possible, but very iffy, considering Japan's total isolation from the West at this time..
Then there are all those lights. As wasabi-chan pointed out on WotC, a large building lit that brightly and all over (see Tsutenkatu Tower frame above) almost certainly has gaslight, which it pretty much can't have prior to 1792 at the earliest (more like sixty years later--the 1792 date is for England, while the first recorded gas lighting in Japan was installed at the Yokohama rail station in the fall of 1872). Of course, if it has gas lighting, it may well also have gas heat, hence steam, hence hydraulic elevators. So I guess the whole building is one great big anachronism--the largest single one in the series. No wonder it's regarded as the best in Osaka.
from Ufiusu:
At the end of Episode 15, Yatsuha says "bye bye." From Dictionary.com:
"No doubt more than one reader has wondered exactly how goodbye is derived from the phrase 'God be with you.'
To understand this, it is helpful to see earlier forms of the expression, such as God be wy you, god b'w'y,
godbwye, god buy' ye, and good-b'wy. The first word of the expression is now good and not God, for good replaced
God by analogy with such expressions as good day, perhaps after people no longer had a clear idea of the original
sense of the expression. A letter of 1573 written by Gabriel Harvey contains the first recorded use of goodbye:
“To requite your gallonde [gallon] of godbwyes, I regive you a pottle of howdyes,” recalling another contraction, "howdy",
that is still used [presumably via "how do ye"/"how do you do"]."
I'm not sure if this counts as a true anachronism, as I'm not 100% sure that it wasn't used, but it is unlikely that
the Japanese would be using "goodbye" (especially "bye bye") in the early years of the Edo period.
From Dakameleon:
*nods* 'bai bai', as the Japanese romaji would spell it, is an expression they've picked up and
added to their modern vocab. It's really only used by youth (i.e. < 30) and in informal situations. It's part
of the whole English-word anachronism I mentioned earlier [see this discussion in notes on Episode 1].
Cultural note: The Japanese really have whole heartedly adopted foreign words. If you compare this with China,
you'll note that there's a huge difference.
I'd say a lot of the foreign words ('gairaigo') come from WWII and after. The biggest exception is probably the
word for telephone - introduced to Japan before the 20th century - [which] is 'Denwa' (= 'electric conversation'),
but since then they've adopted words from German (arubaito = part-time job, from German Arbeit) and god knows how many
from English, due to the American occupation after WWII.
The only one from before that period I can think of is 'pan', for bread (from the Portugese, iirc)
Anachronisms Occurring Only in the English Dub:
--Mugen says "Jackpot!" when he sees the gold. (--Thanks to Kyle Robert Lewis for noting this.) This card-playing term is not recorded anywhere before 1880.