Notice that $n \circledast \frac{1}{2} = \frac{1}{2} \circledast n = n + \frac{1}{2} - 2n \cdot \frac{1}{2} = \frac{1}{2}$. So $A=\left(...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast \frac{3}{2014}\right)...\right)\circledast \frac{2013}{2014}=\left(...\left(...\left(...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast \frac{3}{2014}\right)...\right)\circledast \frac{1007}{2014}\right)\circledast \frac{1008}{2014}...\right)\circledast \frac{2013}{2014}=\left(...\left(...\left(...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast \frac{3}{2014}\right)...\right)\circledast \frac{1}{2}\right)\circledast \frac{1008}{2014}...\right)\circledast \frac{2013}{2014}=\left(...\left(\left(\frac{1}{2}\circledast \frac{1008}{2014}\right)\circledast \frac{1009}{2014}\right)...\right)\circledast \frac{2013}{2014}=\boxed {\frac{1}{2}}$. ...I wonder if any pattern happens with odd denominator >.> probably not tbh