Several identities for the log integral lead with the terms $\log \log n + \gamma$, where $\gamma$ is the Euler–Mascheroni constant.
So, for example, there's the well-known
$$\text{li}(n) = \log \log n + \gamma + \sum_{k=1}^\infty \dfrac{(\log n)^k}{k! k}$$
and
$$\text{li}(n) = \log \log n + \gamma + n^{\frac{1}{2}}\sum_{x=1}^\infty \frac{(-1)^{x-1}(\log n)^x}{x! 2^{x-1}}\sum_{k=0}^{\lfloor (x-1)/2 \rfloor} \frac{1}{2k+1} $$
or (I don't have good references for the rest of these but they all hold empirically), with $L_n(x)$ the Laguerre polynomials,
$$\text{li}(n) = \log \log n + \gamma + \lim_{x \rightarrow 0}\frac{L_{-x}(n)-1}{x}$$
and
$$\text{li}(n) = \log \log n + \gamma + \frac{\partial}{\partial x}L_{-x}(n) \text{ at } x=0$$
and
$$\text{li}(n) = \log \log n + \gamma + \lim_{c \rightarrow 1^+} \sum_{j=1}^{\lfloor \frac{\log n}{\log c}\rfloor}\frac{c^j - 1}{j}$$
and
$$\text{li}(n) = \log \log n + \gamma + \sum_{k=1}^\infty k^{-1}(\frac{\Gamma(k, -\log n)}{\Gamma(k)}-1) $$
where $\Gamma(k,n)$ is the upper incomplete gamma function.
In other instances, such as
$$\text{li}(n)=\int_0^n\frac{dt}{\log t}$$
and
$$\text{li}(n)=-\pi i - \Gamma(0, -\log n)$$
of course, there is no $\log \log n + \gamma$.
Is there a specific reason why so many of these identities lead with $\log \log n + \gamma$? I find it very perplexing.