Maybe like so:

    DIGITPRECISION = 100000;
    ORDER = 30;
    NUMBER = 10^4900;
    sol2 = Table[ ser2 = Normal@Series[Exp[x^2]*Erfc[x], {x, Infinity, k}]; 
    s2 = Block[{$MaxExtraPrecision = Infinity}, N[ser2 /. x -> NUMBER, DIGITPRECISION]]; 
    ANSWER = s2, {k, 1, ORDER}]; m = Last[Counts[sol2]];
    If[TrueQ[m >= 3], "The required precision is obtained", "Not enough ,increase ORDER"]

   (*The required precision is obtained*)

   ANSWER
   (* 5.6418958354...34303*10^-4901*)

Hi,

The example posted by Mariusz exactly touches the problem, showing the overflow. As I wrote, I need to calculate Exp[x^2]*Erfc[x] in the entire range of x corresponding to extended precision variables. The range is up to about 10^4900. Therefore, the values such as x=10^7 or x=10^8 are far too small.

Les?aw

As a power series expansions: Testing:

Block[{$MaxExtraPrecision = Infinity},  N[Exp[x^2]*Erfc[x] /. x -> 10^7, 100]]
(* 5.64189583547753466000161712821652064212873489131706443451983934426010
2089725651175753054274046517361*10^-8 *)

n = 14; (*The result agrees to 100 decimal places*)
ser = Normal@Series[Log[Exp[x^2]*Erfc[x]], {x, Infinity, n}]; s = Block[{$MaxExtraPrecision = Infinity}, 
N[ser /. x -> 10^7, 100]]; Exp[s]
(* 5.64189583547753466000161712821652064212873489131706443451983934426010
208972565117575305427404651736*10^-8 *)

Thanks. Yes, I assume real-valued x. I doubt if expansion of Log[1+y] as a series of y can be useful for any computing, as the series converges extremely slowly. Power series of x, well, yes, the expansions are well known, but my question was more about how to force MATHEMATICA into a truly multiprecision mode of operation in this case. It's also frustrating why there is no special function implemented for this expression. As far as I can see, another similar expression known as the Dawson integral has already been included in MATHEMATICA. So, why not including Exp[x^2]*Erfc[x] ? I might reformulate my question in another way: why one cannot compute Exp[-x] with arbitrary precision for arbitrary x? This appears impossible due to the existing limitation on the range of real values (if I understand the situation). Les?aw