In Maple, series(f(x), x=x0, order) will produce an expansion where the number of terms is specified by "order". But is it possible to specify the maximum power of (x-x0) that is needed, regardless of the starting order? This is especially a problem if the starting order is negative. To make it worse, in practice the number of terms produced by Maple may not be equal to "order", when nontrivial algebraic cancellations change the exact starting order. Experimenting with different choices of "order" case by case is not practical, when automation is needed for many runs involving different functions. MultiSeries:-series doesn't solve my problem, either.
P.S. MultiSeries:-series is more reliable than series in giving the number of terms specified by the user, but it still breaks down sometimes for complicated problems, for example producing much fewer terms than specified. My main concern, i.e. being able to specifiy an absolute maximium power, is also not addressed by switching from series to MultiSeries:-series.