char m_buf[10];

  udp::endpoint m_endpoint;

<-->

  udpSocket.async_receive_from(buffer(buf), m_endpoint, bind(ReadHandler, ...));

  service.run();

If more than 10 characters were in the datagram packet, how can they be accessed?

Calling other read methods (after processing in the ReadHandler) just appears to block (I guess it is looking for another packet).

In the ReadHandler you can synchronously read the remained data , like this (untested!):
size_t avail = udpSocket.available();
std::vector<char> data(avail);
udpSocket.read(buffer(data));