What is a byte? Let's assume the modern eight bits.
What is the representation? What other values have to be possible in that representation?
Suppose we assume a pure binary representation which can handle all of the integers from 0 to 65535. In this representation, 65535 is represented by 16 bits: 1111111111111111. This requires two eight-bit bytes.
If the requirement is to have a representation that only handles the two values { 0, 65535 }, or any other choice of two numbers or symbols, we can encode that in as little as one bit: a fraction of a byte.
If we have to represent all integers from -65535 to 65535, using two's complement (or one's complement will need 17 bits, and that will require three bytes.
We can also think about handling the value 65535.0 in the smallest possible floating-point representation we can think of (which still resembles IEEE). The abstract mantissa corresponds to the binary representation and so requires 16 bits, valued 1. However, in floating-point, the leading 1 can be assumed and so it disappears: every nonzero floating-point value is 1.whatever x (base)^exponent: the 1 is always there, and so we don't have to dedicate a bit to it. Thus the mantissa requires only 15 bits. If we do not allow any exponent in our floating-point representation, only a sign bit, then we are back to 16 bits. But that's a "strawman" floating-point. Genuine floating-point should be able to float the point, so we need an exponent field. Not only that, let's make it a requirement that the exponent must actually encode the exponent value that is required, namely 65535 = 1111111111111111b = 1.111111111111111b x (10b)^1111b, as well as every lower exponent down to zero. Moreover, let's make it a requirement that negative exponents must be handled so we can represent small fractional values closer to zero than 1.0: all exponent values from -16 to +15. That requires a five bit exponent. Thus, in total, we need one bit for the sign, five bits for the exponent and fifteen for the mantissa: 21 bits. That fits into three bytes.