The expansion of a functional around a function is drastically more complicated than suggested in the question and in other answers here so far.
Wheras the Taylor series of $f(x)$ around point $x_0$ is written simply as:
\begin{alignat}{2}
f(x) &= \sum_{n=0}^\infty c_n(x-x_0)^n &~~~,~~~c_n = \frac{f^{(n)}(x_0)}{n!}\tag{1}~,
\end{alignat}
a Volterra series expansion of $f[x(r)]$ around the function $x_0(r)$ is significantly more complicated, as it requires integration over a dummy-variable version of $r$, an additional time for each successive term in the series):
\begin{align}
f[x(r)] &= \sum_{n=0}^\infty \int \int \cdots \int c_n\left(\prod_{i=0}^n \left(x(r)-x_0(r^{(i)})\right) \right) \textrm{d}r^\prime \textrm{d}r^{\prime \prime} \cdots \textrm{d}r^{(n)}\tag{2}\\
&=\int \sum_{n=0}^\infty c_n \prod_{i=0}^n \left(x(r)-x_0(r^{(i)})\right) \textrm{d}r^{(i)}\tag{3}\\
c_n &=\frac{f^{(n)}[x_0(r)]}{n!}\tag{4}\\
&= \frac{1}{n!}\frac{\delta f[x_0(r)]}{\delta f[x(r^\prime)]\delta f[x(r^{\prime\prime})]\cdots \delta f[x(r^{(n)})]}. \tag{5}\\
\end{align}
Eq. 2 has been used in practical applications, for example in this 1994 paper in which $r$ represented a 3-dimensional vector so each integral in Eq. 2 was in fact a triple integral. Chapter 5 of this classical field theory book (PDF) by Nicholas Wheeler also has a lot of information and analogies about the above expansions.
I'll write the expansion out for you one more time in a way that might make it even more clear to you how this expansion compares with the Taylor expansion with which you are probably very familiar:
$$\tag{6}{\small
f[x(r)] = f[x_0(r)] + \int \frac{f^{(1)}[x_0(r)]}{1!}\left(x(r)-x_0(r^\prime\right))\textrm{d}r^\prime + \int\!\!\!\int \frac{f^{(2)}[x_0(r)]}{2!}\left(x(r)-x_0(r^\prime\right))^{(2)}\textrm{d}r^\prime\textrm{d}r{^{\prime\prime}} + \cdots,}
$$
in which:
$$
\left(x(r)-x_0(r^\prime\right))^{(2)}\equiv\left(x(r)-x_0(r^\prime\right))\left(x(r)-x_0(r^{\prime\prime}\right)).
$$
This expansion has been the subject of my first MathOverflow question (earlier today!) which was about how to extend the Volterra series to functionals of more arbitrary complex-valued functions much like the Laurent series extends the concept of a Taylor series to more arbitrary functions of complex variables:
I also answered today my first MathOverflow question, which was a 2.5 year old unanswered question about how to extend the Pade approximant in this way:
Some other Mathematics.SE questions which may interest you are:
