why haven't certain well-researched classes of mathematical object been framed by category theory? Category theory is doing/has done a stellar job on Set, FinSet, Grp, Cob, Vect, cartesian closed categories provide a setting for $\lambda$-calculus, and Baez wrote a paper (Physics, Topology, Logic and Computation: A Rosetta Stone) with Mike Stay about many of the interconnections between them. 
But there are mathematical objects that aren't thought of in a category-theoretic fashion, at least the extant literature doesn't tend to treat them as such. For instance nobody talks about Series, Products, IndefInt as being categories in their own right. (infinite series, infinite products, and indefinite integrals, respectively). (google searches for the phrase "the category of infinite series" in both the web and book databases have no hits whatsoever).  I suppose my question is: why not?
 A: Paul Taylor's "Abstract Stone Duality"
http://www.paultaylor.eu/ASD/
is an attempt to recast elementary real analysis (including sequences)
involving categorical ideas.
A: Fundamentally I agree with Mike Shulman's comment and I do not really want to claim the following fancy language is at all necessary to answer this question, but you may (or may not) find it illuminating.
From the standpoint of higher category theory, categories (i.e., 1-categories) are just one level among many in a family of mathematical structures.  Typically a mathematical object will "naturally" exist as an n-category for some particular n.  For example, Set is naturally a 1-category, while Cat is naturally a 2-category.  Your examples Series and so on seem to just be 0-categories, i.e., sets, since as Pete explained in his answer, there is no obvious natural notion of morphism between infinite series.  Asking why Series is not a 1-category is like asking why Set is not a 2-category; these are just not the natural categorical levels that these objects live at.
A: To turn a class into a category, you need a notion of morphisms between objects in the class.  That's the long and short of it.
Consider for instance the class of infinite real series, say viewed as the set $S = \mathbb{R}^{\aleph_0}$ of sequences of real numbers.  (There is often some notational and "ontological" confusion between the terms of an infinite series, its associated sequence of partial sums, and its sum, if it has one.  Which one of these "is" the series?  But such considerations are not relevant here and indeed are usually viewed as antithetical to the categorical point of view.)  To get a category, you need to identify a set of morphisms between any two elements of this set.  This can certainly be done in any number of ways -- for instance one could use the ordering induced from the standard ordering on $\mathbb{R}$ and the lexicographic ordering of the sequence, and then $S$ is a totally ordered set.  We could then define a category by having $\operatorname{Hom}(s,t)$ to be a one point set if $s \leq t$ and the empty set otherwise (and then take the unique composition law of morphisms, defined when $s \leq t \leq u$).  
But the question is: what does this category have to do with any aspect of the theory of 
infinite series?  Apparently nothing.  You could create any number of other categories with underlying set $S$ but you run into the same problem: the very old and extremely well-developed area of mathematics which studies the convergence and divergence of real infinite series simply does not have anything evident to do with any notion of "morphisms" between infinite series.  
Similarly for the other examples you mention.  Categorical structure is a very fundamental kind of mathematical structure; it's a great way of thinking and unifies and conceptualizes the study of many kinds of mathematical objects in highly disparate fields.  But it doesn't explain everything, and it is frankly a bit weird to think it should.    
A: A Google scholar search for "category theory" "power series" brings up the paper: Elements of Stream Calculus::(An Extensive Exercise in Coinduction). So series can be usefully thought of in a category-theoretic fashion, and although this involves formal series the methods can be used to find solutions to differential equations.
A: For just another point, we could contemplate the hilarious line "why can't a woman... be more like a man", from you-know-where. 
That is, the style of (much) "analysis" reflects as much the personalities of the participants as it does the "mathematical reality"... the latter being self-referentially "defined" by the personalities.
That is, perhaps the issue addressed by the question resides in the implicit assumptions of the questioner? :)
In my own experience, "(naive?) category theory", in the sense of paying attention more to interactions among objects than to their internal structures, has been extremely helpful, at the very least in showing that various seeming "choices" or "constructions" were irrelevant... But I do recognize that other people thing other-ly...
A: I really like to think of $\mathbb{N}$-graded $R$-modules as power series $\bigoplus_{n \in \mathbb{N}} M_n \otimes X^{\otimes n}$, where each "coefficient" $M_n$ is an $R$-module and $X$ is the graded module concentrated degree $1$ and which is $R$ there. Therefore, we have a category of power series, where a morphism is just a family of morphisms between the coefficients. Actually this is a symmetric monoidal category - the tensor product is given by some convolution. And the same works if we replace $\mathsf{Mod}(R)$ by any cocomplete symmetric monoidal category. This is spelled out for example in Section 5.4 of my thesis.
In order to get a connection to power series in analysis, we might endow the unit interval $[0,1]$ with the structure of a cocomplete symmetric monoidal category (cf. Example 3.1.6 in loc.cit.): We use the usual ordering to make it a (thin) category and the usual multiplication to make it a symmetric monoidal category. Colimits are given by suprema. Therefore, we get a cocomplete symmetric monoidal category of sequences valued in $[0,1]$. This is again just an order with a multiplication, where we have $(a_n) \leq (b_n)$ iff $a_n \leq b_n$ for all $n$, and $((a_p) \cdot (b_q))_n = \sup_{p+q=n} a_p \cdot b_q$.
Also notice that coends in category theory capture some ideas of (definite) integration. See MO/78471 for some intuition.
A: It might be worth noting that the problems of computing Feynmann integrals in quantum field theory is one that is traditionally phrased as one of analysis, but is now studied by pure mathematicians using categorical techniques (among others).  
