Expansion of inverse logarithmic integral in terms of lambert w Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form:
$\operatorname{li}^{-1}(n)=n\sum _{i=1} a_i(-1)^{i+1} W_{-1}\left(-e/n\right){}^{2-i} $
where $a_i$ begins: $\small{-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, \dots}$
where the next one $\small{\approx 1465235 + \epsilon, \ \epsilon =-1/12?)}$.
The above should be accurate to $o(1/\log(n)^{10})$ - is this correct? Is there a nice way to obtain the coefficients other than by computation?
For convenience:
f[n_] := With[
{a = {-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, 17582819/12}}, 
n Sum[(-1)^(1 + k) a[[k]] ProductLog[-1, -E/n]^(2 - k), {k, 1, 11}]];
g[n_] := Quiet[x /. FindRoot[LogIntegral@x == n, {x, N[n Log[n], 200]}, 
WorkingPrecision -> 200]];

Abs@Log@N[1/Log[#]^10] &[10^10^7]
Abs@Log@N[1 - f@#/g@#] &[10^10^7]

 A: You may find the following result interesting. The point is to highlight that by reasoning in terms of $p_n -n $ it is possible to find new expressions of the asymptotic trend.
let $p_{n}$ be the nth-prime number and $W_{t}$ the Lambert-W function / $W(z)e^{W(z)}=z $ /:
$ \exists \ \ l \in \mathbb N:\forall n>l$ 
$$ p_{n}<n-nW_{-1}\bigg(\frac{-e^{2}}{n}\bigg)$$
C. AXLER in [1] showed that $ \exists \ \ l \in \mathbb N:\forall n>l$ :

$ p_{n}<n\Big(\ln(n)+\ln_{2}(n)-1+\frac{\ln_{2}(n)-2}{\ln(n)}-\frac{\ln_{2}(n)^{2}-6\ln_{2}(n)+10.667}{2\ln(n)^{2}}\Big)=\Theta_{n}$
$ p_{n}>n\Big(\ln(n)+\ln_{2}(n)-1+\frac{\ln_{2}(n)-2}{\ln(n)}-\frac{\ln_{2}(n)^{2}-6\ln_{2}(n)+10.667}{2\ln(n)^{2}}\Big)=\Phi_{n}  $ 

So
$p_{n}=n\Big(\ln(p_{n}-n)-\ln(p_{n}-n)+\frac{p_{n}}{n}\Big)<n\Big(\ln(p_{n}-n)-\ln(\Phi_{n} -n)+\frac{\Theta_{n}}{n}\Big)$
$j_n=\ln(\Phi_{n} -n)-\frac{\Theta_{n}}{n}$
Since we have $j_n>1 \ $for $n\geq32$ (by computation)
$\ln(p_{n}-n)+\frac{p_{n}}{n}>j_n>1$
$p_{n}<n\Big(\ln(p_{n}-n)-j_n\Big)<n\Big(\ln(p_{n}-n)-1\Big)$.

$p_n+n<n\ln(p_{n}-n)$

$e^{\ \frac{p_n}{n}+1}<p_n-n$
$e^{2}e^{\ \frac{p_n}{n}-1}<n(\frac{p_n}{n}-1)$
$-\frac{e^{2}}{n}>-(\frac{p_n}{n}-1) e^{-(\ \frac{p_n}{n}-1)}$

$$ p_{n}<n-nW_{-1}\bigg(\frac{-e^{2}}{n}\bigg) $$

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[1] NEW ESTIMATES FOR THE n-TH PRIME NUMBER, CHRISTIAN AXLER, Jun 2017 https://arxiv.org/pdf/1706.03651.pdf
https://math.stackexchange.com/questions/3476635/prime-numbers-upper-bound-involving-lambert-w-proof
https://www.researchgate.net/publication/258373882_Primes_and_the_Lambert_W_function
