How many operations at maximum needed to compute powering of a number, i.e. $N^k$?
It depends on what you mean by operations. If multiplication of unbounded numbers is considered one operation, then repeated squaring takes at most $2L$ operations. In that case we don't care about the size of $N$.

If we only consider multiplication of $L$-bit integers as a unit operation, then multiplying $ML$-bit integers takes $O(M^2)$ operations using the high-school algorithm (in fact, much better algorithms are known), and so repeated squaring, in which the integers have size $L,2L,4L,\ldots,2^LL$, takes $O(1^2 + 2^2 + \cdots + 2^{2L}) = O(2^{2L}L)$ operations. Using fancier multiplication algorithms that take $\tilde{O}(L)$ operations (if they exist in this setting) reduces the number of operations to $\tilde{O}(2^L)$.