The simplest methods are usually the best. Try this code
N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15]//InputForm which returns
4.589999999999999999999998802094999`15.*^-25 You can get more digits by increasing the 15 digits precision.
For example, with 34 digits precision the result returned is
4.5899999999999999999999988020950000000000000000001613`34.*^-25