Sure. Anything that maps one state to another is an operator. If $A$ satisfies this definition, namely that when applied to a state it gives you a state, then so does repeated application of $A$.

For example, suppose you have a set of quantum states $\lvert i\rangle$ for various values of $i$, parametrized so that $A\lvert i\rangle = \lvert i+1\rangle$. Then

$$A^2\lvert i\rangle = A(A\lvert i\rangle) = A\lvert i+1\rangle = \lvert i+2\rangle$$

Hopefully you can see how this generalizes, so that $A^2$ is the operator that takes $\lvert i\rangle\to\lvert i+2\rangle$.

Sure. Anything that maps one state to another is an operator. If $A$ satisfies this definition, namely that when applied to a state it gives you a state, then so does repeated application of $A$.

For example, suppose you have a set of quantum states $\lvert i\rangle$ for various values of $i$, parametrized so that $A\lvert i\rangle = \lvert i+1\rangle$. Then

$$A^2\lvert i\rangle = A(A\lvert i\rangle) = A\lvert i+1\rangle = \lvert i+2\rangle$$

Hopefully you can see how this generalizes, so that $A^2$ is the operator that takes $\lvert i\rangle\to\lvert i+2\rangle$.

And yes, you can generalize this to construct an operator as a power series of other operators. This is how the exponential of an operator is defined, for example.