What is the maximum number of bytes for a UTF-8 encoded character?
The maximum number of bytes per character is 4 according to RFC3629 which limited the character table to U+10FFFF:
In UTF-8, characters from the U+0000..U+10FFFF range (the UTF-16 accessible range) are encoded using sequences of 1 to 4 octets.
(The original specification allowed for up to six byte character codes for code points past U+10FFFF.)
Characters with a code less than 128 will require 1 byte only, and the next 1920 character codes require 2 bytes only. Unless you are working with an esoteric language, multiplying the character count by 4 will be a significant overestimation.
Without further context, I would say that the maximum number of bytes for a character in UTF-8 is
answer: 6 bytes
The author of the accepted answer correctly pointed this out as the "original specification". That was valid through RFC-2279 1. As J. Cocoe pointed out in the comments below, this changed in 2003 with RFC-3629 2, which limits UTF-8 to encoding for 21 bits, which can be handled with the encoding scheme using four bytes.
answer if covering all unicode: 4 bytes
But, in Java <= v7, they talk about a 3-byte maximum for representing unicode with UTF-8? That's because the original unicode specification only defined the basic multi-lingual plane (BMP), i.e. it is an older version of unicode, or subset of modern unicode. So
answer if representing only original unicode, the BMP: 3 bytes
But, the OP talks about going the other way. Not from characters to UTF-8 bytes, but from UTF-8 bytes to a "String" of bytes representation. Perhaps the author of the accepted answer got that from the context of the question, but this is not necessarily obvious, so may confuse the casual reader of this question.
Going from UTF-8 to native encoding, we have to look at how the "String" is implemented. Some languages, like Python >= 3 will represent each character with integer code points, which allows for 4 bytes per character = 32 bits to cover the 21 we need for unicode, with some waste. Why not exactly 21 bits? Because things are faster when they are byte-aligned. Some languages like Python <= 2 and Java represent characters using a UTF-16 encoding, which means that they have to use surrogate pairs to represent extended unicode (not BMP). Either way that's still 4 bytes maximum.
answer if going UTF-8 -> native encoding: 4 bytes
So, final conclusion, 4 is the most common right answer, so we got it right. But, mileage could vary.