Careful, we don't actually have
$\log(n)=\sum\limits _{k=1}^{\infty}\sum\limits _{a=1}^{n-1}\frac{1}{kn-a}-\sum\limits _{k=1}^{\infty}\frac{n-1}{kn},$
since neither series converges. Instead we have
$\log(n)=\lim_{M\rightarrow\infty}\sum\limits _{k=1}^{M}\sum\limits _{a=1}^{n-1}\frac{1}{kn-a}-\sum\limits _{k=1}^{M}\frac{n-1}{kn}.$
Now, since
$\sum\limits _{k=1}^{M}\frac{n-1}{kn}=\sum\limits _{k=1}^{M}\frac{1}{k}-\sum\limits _{k=1}^{M}\frac{1}{kn}$
we can rewrite
$\log(n)=\lim_{M\rightarrow\infty}\sum\limits _{k=1}^{M}\sum\limits _{a=0}^{n-1}\frac{1}{kn-a}-\sum\limits _{k=1}^{M}\frac{1}{k}=\lim_{M\rightarrow\infty}\sum_{k=1}^{nM}\frac{1}{k}-\sum_{k=1}^{M}\frac{1}{k}.$
Thus we have
$\gamma=\lim_{n\rightarrow\infty}\lim_{M\rightarrow\infty}\left(\sum_{k=1}^{n}\frac{1}{k}-\sum_{k=1}^{nM}\frac{1}{k}+\sum_{k=1}^{M}\frac{1}{k}\right).$
Personally, I quite like this limit since it has a nice symmetry when we switch the order of the limits. Also, it generalizes to give limits over l variables which are invariant under permutation. Let $H_{k}$ be the $k^{th}$ harmonic number. Then the above was
$\gamma=\lim_{n\rightarrow\infty}\lim_{M\rightarrow\infty}\left(H_{n}-H_{nm}+H_{m}\right).$
Here is $l=3$:
$\gamma=\lim_{n_{1}\rightarrow\infty}\lim_{n_{2}\rightarrow\infty}\lim_{n_{3}\rightarrow\infty}\left(\left(H_{n_{1}}+H_{n_{2}}+H_{n_{3}}\right)-\left(H_{n_{1}n_{2}}+H_{n_{2}n_{3}}+H_{n_{3}n_{1}}\right)+\left(H_{n_{1}n_{2}n_{3}}\right)\right).$
In general
$\gamma=\lim_{n_{1}\rightarrow\infty}\cdots\lim_{n_{l}\rightarrow\infty}\left(\sum_{i_{1}=1}^{l}H_{n_{i_{1}}}-\sum_{i_{1}
So I guess it depends on what you mean by “simplify further.” I find the form
$\gamma=\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}\left(H_{n}-H_{nm}+H_{m}\right)$
to be quite simple.
Hope that helps,