MATL, 14 bytes

1111G:"t*V4:)U

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MATL, 14, 13 bytes

1111G:"UV4:)U

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Explanation:

1111            % Push 1111
    G           % Push input
     :"         % Input times:
       U        %   Square the top of the stack
        V       %   Convert it to a string
         4:)    %   Take the first four digits
            U   %   Convert it back to a number
                % Implictly display