Let N be the fifth largest number that can be created by combining 2021 1's using addition, multiplication, and exponentiation, in any order (parentheses are allowed). If f(x)=\log _{2}(x), and k is the least positive integer such that f^{k}(N) is not a power of 2, what is the value of f^{k}(N)?
(Note: f^{k}(N)=f(f(\cdots(f(N)))), where f is applied k times.)