For any $n$, is there a prime factor of $2^n-1$ which is not a factor of $2^m-1$ for $m < n$? Is it guaranteed that there will be some $p$ such that  $p\mid2^n-1$ but $p\nmid 2^m-1$ for any $m<n$?
In other words, does each $2^x-1$ introduce a new prime factor?

No. $2^6-1 = 3^2 \cdot 7$. But we have that $3|2^2-1$ and $7|2^3-1$
