A variation of Michael E2's answer that uses O instead of Series and thus avoids its bug.
A = a + c (d - a b d) + e c (a + b) f + b - 1;
Normal[(A /. v : Alternatives @@ Variables[A] :> v $t) +
O[$t]^2] /. $t -> 1
-1 + a + b
azerbajdzan
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