This OPE is not trivial, it just doesn't have singular terms.
Suppose there is a quasi-primary operator $\mathcal{O}$ with weight $(h,\bar h)$ appearing in the right-hand side. We can compute the coefficient with which it appears by looking at three point function
$$
\langle T(z)\bar T(\bar w)\mathcal{O}(x,\bar x)\rangle=\frac{f_{T\bar T\mathcal{O}}}{(z-w)^{2-h}(\bar z-\bar w)^{2-\bar h}(x-w)^{h-2}(\bar x-\bar w)^{\bar h+2}(z-x)^{2+h}(\bar z-\bar x)^{\bar h-2}}.
$$
The right-hand side is fixed by global conformal invariance up to the coefficient $f_{T\bar T\mathcal{O}}$. However, the left-hand side only depends on $z$ and not $\bar z$, so we must conclude $\bar h=2$. Similarly because it only depends on $\bar w$ and not $w$ we must conclude $h=2$. This means that no singular terms can appear in the OPE because these must necessarily have $h+\bar h<2$. But then we can define the operator
$$
(T\bar T)(z,\bar z) \equiv T(z)\bar T(\bar z).
$$
It is a quasi-primary and has dimensions $(h,\bar h)=(2,2)$. It is in fact the only quasi-primary that appears in the OPE. The OPE takes the form simply
$$
T(z)\bar T(\bar w) = (T\bar T)(z,\bar w)=\sum_{n=0}^\infty \frac{1}{n!}(z-w)^n\partial^n_w(T\bar T)(w,\bar w).
$$
The operator $T\bar T$ can in fact be defined in any 2d QFT, not necessarily conformal, but the argument is more subtle. Presently there is a great deal of research into the theories one gets by adding $T\bar T$ to the Lagrangian. (Note that this is an irrelevant deformation.) Search for "$T\bar T$-deformation".
Added: Per request of the OP, here is a crash course in OPEs in conformal field theories. Since this answer only requires global conformal invariance, I will not discuss implications of Virasoro symmetry. Because of this, the below applies (with small modifications to accommodate general spin) in CFTs in $d\geq 2$. Virasoro symmetry also leads to straightforward modifications.
Below $x_i$ denote space-time points.
Any CFT possesses operator product expansion that is convergent in vacuum state. That is,
$$
\mathcal{O}_1(x_1)\mathcal{O}_1(x_2)|0\rangle=\sum_i f_{\mathcal{O}_1\mathcal{O}_2\mathcal{O}_i} C_{12i}(x_1,x_2,x_3,\partial_{x_3})\mathcal{O}(x_3)|0\rangle.
$$
Point $x_3$ is in principle arbitrary and often taken to be $x_3=x_2$. Here the differential operator $C_{12i}(x_1,x_2,x_3,\partial_{x_3})$ is completely fixed by conformal symmetry. It depends only on the quantum numbers of opeartors $\mathcal{O}_1,\mathcal{O}_2,\mathcal{O}_i$. The coefficient $f_{\mathcal{O}_1\mathcal{O}_2\mathcal{O}_i}$ is not fixed by conformal symmetry and represents the dynamical information about the theory.
This expansion is exact and covergent. It is often written by omitting the vacuum state $|0\rangle$. This is because it is often used inside Euclidean correlation functions, where one doesn't necessarily have to talk about a particular quantization. In Euclidean correlation functions one interprets vacuum state in radial quantization around point $x_3$. The OPE is applicable in a Euclidean $n$-point correlation function if there exists a sphere around $x_3$ which only contains the operators $\mathcal{O}_1,\mathcal{O}_2$ at $x_1$ and $x_2$ and no other operators.
One can compute the coefficient $f_{\mathcal{O}_1\mathcal{O}_2\mathcal{O}_i}$ by looking at three-point function $\langle\mathcal{O}_1\mathcal{O}_2\mathcal{O}_i\rangle$ and using the OPE inside the three-point function. Since the two-point functions are canonically chosen to be diagonal $\langle\mathcal{O}_i\mathcal{O}_j\rangle\propto \delta_{i,j}$, we have
$$
\langle\mathcal{O}_1(x_1)\mathcal{O}_2(x_2)\mathcal{O}_i(x_3)\rangle=f_{\mathcal{O}_1\mathcal{O}_2\mathcal{O}_i} C_{12i}(x_1,x_2,x'_3,\partial_{x'_3})\langle\mathcal{O}_i(x'_3)\mathcal{O}_i(x_3)\rangle.
$$
Again, often one uses $x'_3=x_2$. Since $C_{12i}(x_1,x_2,x'_3,\partial_{x'_3})\langle\mathcal{O}(x'_3)\mathcal{O}(x_3)\rangle$ is fixed by conformal symmetry and canonical normalization of two-pt functions, the coefficient $f_{\mathcal{O}_1\mathcal{O}_2\mathcal{O}_i}$ is computed by three-point functions. However, this coefficient appears in the OPE, and the OPE is applicable in all $n$-point correlation functions, so there is no lack of generality the OP seems to be worried about.