[Edit: The lemma was revised and proved, changed the point of view from series to sequences.]
[2nd Edit: The proof of the lemma was improved, and now the argument can show that Cauchy series such as $x_1+(x_{1^3+1}^2+\dots x_{2^3}^2)+(x_{2^3+1}^{3}+\dots+x_{3^3}^3)+\dots$, diverge in the $\mathfrak{m}$-adic topology.]
Here is a construction of a Cauchy sequence which does not converge. It is based on the following lemma, a proof of which will is given at the end.
Lemma. Let $\mathfrak{m}_c$ denote the maximal ideal of $K[[x_1,\dots,x_c]]$.
Then there exist sequences of natural numbers $(r_n)_{n\in\mathbb{N}}$, $(c_n)_{n\in\mathbb{N}}$ and a sequence of elements $(p_n\in(\mathfrak{m}_{c_n})^n)_{n\in\mathbb{N}}$ such that:
- $\limsup r_n=\infty$ and
- $p_n$ cannot be written as a sum of $r_n$ terms $\sum_{i=1}^{r_n} a_{i} b_i$ with $a_{i},b_i\in\mathfrak{m}_{c_n}$.
In fact, one can take $c_n=n^2$, $p_n=x_1^{n}+\dots+x_{n^2}^{n}$ and let $r_n=\lceil \frac{n^2}{2(n-1)}\rceil-1$ if $\mathrm{char}\,K\nmid n$ and $r_n=1$ when $\mathrm{char}\,K\mid n$.
It would be more convenient to replace $K[[x_1,x_2,\dots]]$ with the isomorphic ring $$S:=K[[y_{ij}\,|\,i,j\in \mathbb{N}]].$$
For every $n$, there is a ring homomorphism $\phi_n:S\to K[[x_1,\dots,x_{c_n}]]$ specializing $y_{n1},\dots,y_{n{c_n}}$ to $x_1,\dots,x_{c_n}$ and the rest of the variables to $0$.
Let $f_n=p_n(y_{n1},\dots,y_{nc_n})$ and define in $S$ the (formal) partial sums
$$
g_t:=\sum_{n=t}^\infty f_n.
$$
Then, by construction, $(g_t)_{t\in\mathbb{N}}$ is a Cauchy sequence relative to the $\mathfrak{m}$-adic topology, but it does not converge.
Indeed, if $(g_t)_t$ converges in the $\mathfrak{m}$-adic topology, then it also converges relative to the (coarser) grading topology of $S$, and so must converge to $0$. However, if this is so, there exists $t\in\mathbb{N}$ such that $g_t=\sum_{n=t}^\infty f_n\in \mathfrak{m}^2$. In particular, we can write $\sum_{n=t}^\infty f_n=\sum_{i=1}^u a_{i}b_i$ with $a_{i},b_i\in \mathfrak{m}$.
Choose $n\geq t$ sufficiently large to have $r_n\geq u$. Applying $\phi_n$ to both sides of the last equality gives $p_n=\sum_{i=1}^u (\phi a_{i})(\phi b_{i})$ with $\phi a_{i},\phi b_i\in\mathfrak{m}_{c_n}$, which is impossible by the way we chose $p_n$.
Back to the lemma: Given $f\in K[x_1,\dots,x_c]$, let $D_if$ denote its (formal) derivative relative to $x_i$. The lemma follows from the following general proposition.
Proposition. Let $f\in K[[x_1,\dots,x_c]]$ be a homogeneous polynomial of degree $n$, and let $r\in\mathbb{N}$ denote the minimal integer such that $f$ can written as $\sum_{i=1}^ra_ib_i$ with $a_i,b_i\in \mathfrak{m}_c$. Suppose that $D_1f,\dots,D_cf$ have no common zero beside the zero vector over the algebraic closure of $K$. Then $r\geq \frac{c}{2(n-1)}$.
Proof. Suppose otherwise, namely, that $f=\sum_{i=1}^r a_ib_i$ with $2r(n-1)<c$. Notice that $a_i,b_i$ are a priori not polynomials --- rather, they are power series. To fix that, we write each $a_i$ and $b_i$ as a sum of their homogenous components and rewrite the degree-$n$ homogeneous component of product $a_ib_i$ as a sum of
the relevant components of $a_i$ and $b_i$. By doing this, we see that $f$ can written as $\sum_{i=1}^{r(n-1)}a'_ib'_i$ with $a'_i,b'_i$ being homogeneous polynomials in $\mathfrak{m}_c$.
Let $V$ denote the affine subvariety of $\mathbb{A}^c_{\overline{K}}$
determined by the $2r(n-1)$ equations $a'_1=b'_1=a'_2=b'_2=\dots=0$. It is well-known (see Harshorne's "Algebraic Geometry", p. 48) that every irreducible component of $V$ has dimension at least $c-2r(n-1)>0$. Furthermore, $V$ is nonempty because it contains the zero vector (because $a'_1,b'_1,a'_2,b'_2,\dots\in \mathfrak{m}_c$). Thus, there exists a nonzero $v\in \overline{K}^c$ annihilating $a'_1,b'_1,a'_2,b'_2,\dots$.
Now, by Leibniz's rule, we have $$D_jf=\sum_i D_j(a'_ib'_i)=\sum_i(D_ja'_i\cdot b'_i + a'_i\cdot D_j b'_i).$$
It follows that $v$ above annihilates all the derivatives $D_1f,\dots,D_cf$, a contradiction! $\square$
If it weren't for the passage from power series to polynomials, the proof would work for non-homogenous polynomials and also give the better bound $r\geq \frac{c}{2}$. [Edit: This is in fact possible, see the comments.]
Such a bound would suffice to prove that the Cauchy series $x_1+x_2^2+x_3^3+\dots$ suggested in the question diverges in the $\mathfrak{m}$-adic topology.
