Harlan J. Brothers's approximation to $ e $ ad infinitum? Consider the series generated by Harlan J. Brothers's method for the number $e$
http://en.wikipedia.org/wiki/List_of_representations_of_e
Can they be improved again and again or is there a limit so there exists a BEST approximation to $e$?
 A: It depends on what you mean by "improved" and  "BEST."  The paper @Martin referenced discusses "an algebraic assessment based on the decimal place accuracy (d.p.a.) that is gained from a given number of terms; the greater the d.p.a., the faster the series converges."  This is different from the run-time cost of the computation for which a smaller number of processor operations indicates that the algorithm is inherently more efficient.
While the combination of T terms results in new "compressed" terms containing a polynomial of degree T-1, the algorithm appears reasonably linear (to at least T=100).  Computing 30,000 terms with Newton's method achieves 121,287 correct digits.  Obtaining the same accuracy by computing 3,000 terms with t=10 requires very close to 1/10 the time - it's 9.9 times as fast (these computations are done in Mathematica 9).  Computing just 300 terms with T=100 is 96 times as fast.  However, for T=1000, things slow down - 30 terms are only 705 times as fast.  To my knowledge, there has been no evaluation of the true computational efficiency of the approach.
Personally, I find value in the aesthetic appeal of expressions resulting from combinations that use low values of T.  Here is a more comprehensive list of approximations that supplements the paper:
http://www.brotherstechnology.com/math/e-formulas.html
In short, Newton's series can indeed be algebraically improved again and again. The extent to which such "improvements" can continue to be of practical value is a different matter.  For proven efficiency, you can reference the work of the computational wizards cited here:
http://en.wikipedia.org/wiki/Binary_splitting
