So I was wondering if this is complete power set of the power set of natural numbers.
All sets of any finite combinations of natural numbers that can exist. For example, {1, 2, 4}, {4, 2, 5, 6, 7}, {1000, 1 , 78} etc.
An empty set(Φ) and the set containing all natural numbers(N).
All sets which contain all natural numbers except a particular finite set of natural numbers. For example N - {1, 2, 4}, N - {4, 2, 5, 6, 7}, N - {1000, 1 , 78} etc.
This is all I was able to think of. I feel like I am definitely missing something here.
You've missed the bi-infinite sets - sets which are infinite and have infinite complement. For example, the set of even numbers is not on your list; neither is the set of prime numbers, the set of perfect cubes, or so on.
In fact, most sets of natural numbers are bi-infinite in the precise sense that the set of non-bi-infinite sets of naturals is countable but the set of bi-infinite sets of naturals is uncountable. The first of these claims is a good exercise; the second is Cantor's diagonal argument.