For $u,x>0$, let $P$ be the function given by $$P(x)=\int_0^x\int_u^{u+1}\frac1{\ln t}\,dt\,du\tag1.$$
Is there a closed form for the positive root of $P(x)$, denoted by $\nu$? Can it be written in terms of other constants that may not have a closed form?
Has this constant been studied before? (if so references would be great)
The logarithmic integral is given by ${\rm{li}}(x)$, the integral of $(\ln x)^{-1}$. Its only positive root leads to the Ramanujan-Soldner constant, denoted by $\mu\approx1.4153$. We can write $(1)$ as $$P(x)=\int_0^x{\rm{li}}(u+1)-{\rm{li}}(u)\,du\tag2$$ and since the integral of ${\rm{li}}(u)$ is $u\,{\rm{li}}(u)-{\rm{li}}(u^2)$, we can write $(2)$ as $$P(x)=\left[(u+1)\,{\rm{li}}(u+1)-{\rm{li}}((u+1)^2)-u\,{\rm{li}}(u)+{\rm{li}}(u^2)\right]_0^x\tag{3}$$ so solving $P(x)=0$ is equivalent to finding a constant $\nu$ such that
$$(\nu+1)\,{\rm{li}}(\nu+1)-{\rm{li}}((\nu+1)^2)-\nu\,{\rm{li}}(\nu)+{\rm{li}}(\nu^2)=-\ln2\tag4$$
as $$\lim_{x\to0}(x+1)\,{\rm{li}}(x+1)-{\rm{li}}((x+1)^2)=-\ln2.$$
From W|A an approximation of the constant (link) is $$\nu\approx0.842920299.$$ I am not sure whether there is a closed form, but there may be some simplifications to the LHS of $(4)$ as it is just a combination of the same integrand $(\ln t)^{-1}$ evaluated at different upper limits (for example $\nu+1$ and $\nu^2$). Of course it is possible to change from $\rm{li}(\cdot)$ to $\rm{Ei}(\cdot)$ but I doubt that it is useful.