Normal Order of Normal Order In the first volume of Polchinski page 39 we can read a compact formula to perform normal-order for bosonic fields
$$
:\cal F:~=~\underbrace{\exp\left\{\frac{α'}{4}∫\mathrm{d}^2z\mathrm{d}^2w\log|z-w|^2\frac{δ}{δφ(z,\bar z)}\frac{δ}{δφ(w,\bar zw)} \right\}}_{:=\mathcal{O}}\cal F, \tag{1}
$$
What I do not understand it is that I would like to have (bearing in mind the definition involving $a$ and $a^†$
$$
::\cal F::~=~:\cal F:\tag{2}
$$
but with this formula
$$
\cal O^2\cal F~≠~\cal O \cal F.\tag{3}
$$
EXAMPLE:
$$
:φ(z)φ(w):~=~φ(z)φ(w)-\frac{α'}{2}\log|z-w|^2\tag{4}
$$
but
$$\begin{align}
::φ(z)φ(w)::~=~&:φ(z)φ(w):-\frac{α'}{2}\log|z-w|^2\cr
~=~&φ(z)φ(w)-α'\log|z-w|^2.\end{align}\tag{5}
$$
 A: *

*Short explanation: Polchinski's eq. (1) is not a formula that transforms no normal order into normal order: The expression ${\cal F}$ on the right-hand side of eq. (1) is implicitly assumed to be radially ordered. In fact, eq. (1) is a Wick theorem for changing radial order into normal order, cf. e.g. this Phys.SE post.


*Longer explanation:   When dealing with non-commutative operators, say $\hat{X}$ and $\hat{P}$, the "function of operators" $f(\hat{X},\hat{P})$ does not make sense unless one specifies an operator ordering prescription  (such as, e.g., radial ordering, time-ordering, Wick/normal ordering, Weyl/symmetric ordering, etc.). A more rigorous way is to introduce a correspondence map  $$\begin{array}{c} \text{Symbols/Functions}\cr\cr 
\updownarrow\cr\cr\text{Operators}\end{array}\tag{A}$$
(E.g. the correspondence map from Weyl symbols to operators is explained in this Phys.SE post.)
To define an operator $\hat{\cal O}$ on operators, one often give the corresponding operator ${\cal O}$ on symbols/functions, i.e.,
$$ \begin{array}{ccc} \text{Radial-Ordered Symbols/Fcts}&\stackrel{\cal O}{\longrightarrow} & \text{Normal-Ordered Symbols/Fcts} \cr\cr 
\updownarrow &&\updownarrow\cr\cr
\text{Radial-Ordered Operators}&\stackrel{\hat{\cal O}}{\longrightarrow} & \text{Normal-Ordered Operators}\end{array}\tag{B}$$
E.g. Polchinski's differential operator ${\cal O}$ does strictly speaking only make sense if it acts on symbols/functions. The identification (A) of symbols and operators is implicitly implied in Polchinski.


*Concerning idempotency of normal ordering, see also e.g. this related Phys.SE post.
