For $k=1$ and $m$ a not too small prime, there are fewer than $f=m/(\log_2 m)$ prime factors of $2^m-1$, all of them at least as large as $m$, and your ratio is $O((m/(m-2))^f)$, which is bounded, so the answer is yes the lim inf is finite for $k=1$.

For $k \gt 1$ fixed, we have no nice estimates on the number or size of distinct prime factors of either numerator or denominator of your fraction, and at present we don't have a good understanding of the dynamics of the $\sigma$ or the $\phi$ functions to give a good guess. I am looking at a concept with the working title "factoral abundance" to address questions like this. Getting some help with the estimates raised in How to get a good upper bound on $\sum_{1 \lt d, d|n} \phi(d)/\log d$? would push us nearer to such answers.

Gerhard "And Help On Other Questions" Paseman, 2015.12.29

For $k=1$ and $m$ a not too small prime, there are fewer than $f=m/(\log_2 m)$ prime factors of $2^m-1$, all of them at least as large as $m$, and your ratio is $O((m/(m-2))^f)$, which is bounded, so the answer is yes the lim inf is finite for $k=1$.

For $k \gt 1$ fixed, we have no nice estimates on the number or size of distinct prime factors of either numerator or denominator of your fraction, and at present we don't have a good understanding of the dynamics of the $\sigma$ or the $\phi$ functions to give a good guess. I am looking at a concept with the working title "factoral abundance" to address questions like this. Getting some help with the estimates raised in How to get a good upper bound on $\sum_{1 \lt d, d|n} \phi(d)/\log d$? would push us nearer to such answers.

Gerhard "And Help On Other Questions" Paseman, 2015.12.29

For $k=1$ and $m$ a not too small prime, there are fewer than $f=m/(\log_2 m)$ prime factors of $2^m-1$, all of them at least as large as $m$, and your ratio is $O((m/(m-2))^f)$, which is bounded, so the answer is yes the lim inf is finite for $k=1$.

For $k \gt 1$ fixed, we have no nice estimates on the number or size of distinct prime factors of either numerator or denominator of your fraction, and at present we don't have a good understanding of the dynamics of the $\sigma$ or the $\phi$ functions to give a good guess. I am looking at a concept with the working title "factoral abundance" to address questions like this. Getting some help with How to get a good upper bound on $\sum_{1 \lt d, d|n} \phi(d)/\log d$? would push us nearer to such answers.

Gerhard "And Help On Other Questions" Paseman, 2015.12.29

For $k=1$ and $m$ a not too small prime, there are fewer than $f=m/(\log_2 m)$ prime factors of $2^m-1$, all of them at least as large as $m$, and your ratio is $O((m/(m-2))^f)$, which is bounded, so the answer is yes the lim inf is finite for $k=1$.

For $k \gt 1$ fixed, we have no nice estimates on the number or size of distinct prime factors of either numerator or denominator of your fraction, and at present we don't have a good understanding of the dynamics of the $\sigma$ or the $\phi$ functions to give a good guess. I am looking at a concept with the working title "factoral abundance" to address questions like this. Getting some help with the estimates raised in How to get a good upper bound on $\sum_{1 \lt d, d|n} \phi(d)/\log d$? would push us nearer to such answers.

Gerhard "And Help On Other Questions" Paseman, 2015.12.29