Can one construct a new operator in terms of the powers of another operator? Suppose we have a quantum state, well described by its time-independent wave function Psi. And we have a well-defined Hermitian (self-adjoint) operator $A$. We successfully evaluate the expectation value of the operator $A$. Next we derive the general formula for the higher moments of $A$ (i.e. the expectation value of $A^n$ for $n=2,3,4\ldots $).
In this situation, is it permitted to regard each of the $A^n$ for $n=1,2,3,\ldots$ as a proper operator by itself? 
For example, should every $A^n$ have a positive variance and other statistical properties (as long as we restrict ourselves to the state $\Psi$)? 
Can one make linear combinations of different powers to construct a new operator, e.g. $B = A + A^2$?
Is it allowed to construct new operators acting on $\Psi$, that are defined in terms of their series expansion in $A^n$? For example, $D = \exp(CA)$ where $C$ is a constant?
 A: Most of your question is unclear to me, but I the answer to what I think is the core of your question is yes:


*

*For any hermitian operator $A$ and any well-behaved-enough function $f:\mathbb R\to\mathbb R$, it is possible to construct a new operator $f(A)$ which acts in essentially all important respects as the action of $f$ on $A$.


There are different ways to construct $f(A)$, and they depend on exactly what $f$ is, what "well-behaved-enough" means, and how nice $A$ is. In general, the keyword to search for is function of an operator.
For example, if $f$ is analytic in a large enough region - one that includes all the spectrum of $A$, for example, then  you can define it as
$$
f(A)=\sum_{n=0}^\infty \frac{f^{(n)}(a_0)}{n!}(A-a_0)^n.
$$
Note, in particular, that this includes functions of the form $A+A^2$, which are perfectly allowed. If $A$ is a linear operator then so is $A^2$, and adding two linear operators is bread and butter in linear algebra. There are a few caveats - for example, if the domain of $A$ is smaller than the Hilbert space then the domain of $A^2$ will typically be smaller, so the sum only makes sense in that restricted domain - but this is only the standard measure of care one needs to take when infinite-dimensional spaces are involved.
Alternatively, if $A$ has an eigenvector expansion as $A=\sum_k a_k|a_k\rangle\langle a_k|$, then you can define
$$f(A)=\sum_k f(a_k)|a_k\rangle\langle a_k|.$$
If everything behaves well, then both definitions will match.
Finally, note that one should keep an eye on the dimensional analysis of the whole thing. If $A=x$ has dimensions of position, then it does not make sense to add $x+x^2$, any more than it does to do this in classical mechanics. This is particularly the case with, say, exponentials of operators, like the displacement operator
$$
e^{ix_0\hat p}=\sum_{n=0}^\infty i^n\frac{x_0^n}{n!}\hat p^n
$$
which only makes sense in units where $\hbar=1$. (Otherwise, you need to add in the $\hbar$ explicitly, as $\exp(ix_0\hat p/\hbar$.)
A: 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.
A: For a continuous (linear) operator $A:H\to H$, aka. bounded operator, it is always possible to construct well-defined powers $A^2$, $A^3$, $\ldots.$ Here $H$ denotes a (complex) Hilbert space.
However, the situation changes drastically for a general unbounded operator $A$. Unbounded operators often appear in quantum mechanics, see e.g. this, this and this Phys.SE posts. 
The domain $D(A)\subsetneq H$ of an unbounded operator is never the full Hilbert space! Therefore if the image ${\rm Im}(A)\subseteq H$ is not a subset of the domain $D(A)$, then the square operator $A^2$ does not necessarily make sense on the full domain $D(A)$ of $A$. In other words, the domain $D(A^2)$ of the square operator $A^2$ is in general different from the domain of $A$! Similar for higher powers of $A$.
The topic of unbounded operators is a huge subject in functional analysis, which is impossible to cover in a single Phys.SE post. For a student of operator theory, the natural next couple of questions to ponder is:


*

*Is it possible to extend a domain $D$ of an unbounded operator $A$? 

*If yes, is there a natural way to partially order the set of all possible domains of an unbounded operator $A$?

*Is there a canonical choice of a domain for an unbounded operator $A$?
