The OP asks where this recursion relation might appear in a research context. It appears as a discretization of the <A HREF="http://eqworld.ipmnet.ru/en/solutions/ode/ode0302.pdf">Emden–Fowler nonlinear differential equation</A>,
$$f''(t)=t^{p}[f(t)]^q,$$
for $p=1$, $q=-1$. A particular solution is
$$f(t)=\lambda t^{(p+2)/(1-q)},\;\;\lambda=\left[\frac{(p+2)(p+q+1)}{(q-1)^2}\right]^{1/(q-1)}.$$
One readily checks that the asymptotic limit $x_n\rightarrow \sqrt{3n}$ for $x_n=f'(n)$ is obtained for $p=1$, $q=-1$.

**Conjecture:** The recursion relation
$$x_{n+1}=x_n+n^{p}(x_1+x_2+\cdots+x_n)^q$$
has for $-1-p<q<1$ the limit
$$\lim_{n\rightarrow\infty}n^{(1-q)/(1+p+q)}x_n=\frac{p+2}{1-q}\left[\frac{(p+2)(p+q+1)}{(q-1)^2}\right]^{1/(q-1)}.$$

For $q=-1$ this is stated as an open problem on page 11 of <A HREF="https://link.springer.com/chapter/10.1007/978-3-030-77139-3_1">Sequences of Real Numbers</A>, by Sîntămărian & Furdui.