As you could read on the Documentation page for the "General::ovfl"
message you got, the problem here possible in bad starting values chosen by NonlinearModelFit
. Let us try to specify better starting values:
nlm = NonlinearModelFit[data, a*(1 - Exp[-b*x]), {{a, -1}, {b, 0}}, x];
nlm["BestFitParameters"]
(*=> {a -> -134.847, b -> -0.0101706} *)
Right answer and no error messages.