Is there some really powerful summation method? I know that Borel summation gives you a value if the coefficients are bounded by $n!C^n$.
 Is there a more powerful summation method (with nice properties comparable to Borel) that sums series with coefficients bounded by $(kn)!$ or even $(n^k)!$ ?
 A: In many situations “the most powerful possible generalization” of Borel’s method is applicable.  It was introduced in (zillions of) Écalle’s papers (for example, it is discussed in Chapter 3 of his “NATO” paper).
(Unfortunately, as Écalle’s papers go, they all seem to be “popular expositions” of a [non-existent — or existing in a Platonic sense only] fundamental giga-mega-treatise.  This is not surprising: in its entirety, his work seems to be based on half-a-hundred of very deep non-trivial and intertwined ideas.  It is hard to imagine how to put the tangle of these ideas on paper.)
If all you are interested in is summation, Balser’s books may be a better source.  He untangled a small chunk of these ideas: see the notion of “multisummability”.  (I learned it from the book Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations; I can see other people recommending From Divergent Power Series to Analytic Functions: Theory and Application of Multisummable Power Series as well.
